Old page wikitext, before the edit (old_wikitext) | '[[File:Cobordism.svg|thumb|A cobordism (''W''; ''M'', ''N'').]]
In [[mathematics]], '''cobordism''' is a fundamental [[equivalence relation]] on the class of [[compact space|compact]] [[manifold]]s of the same dimension, set up using the concept of the [[boundary (topology)|boundary]] (French ''[[wikt:bord#French|bord]]'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their [[disjoint union]] is the ''boundary'' of a compact manifold one dimension higher.
The boundary of an (''n'' + 1)-dimensional [[manifold]] ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by [[René Thom]] for [[smooth manifold]]s (i.e., differentiable), but there are now also versions for
[[Piecewise linear manifold|piecewise linear]] and [[topological manifold]]s.
A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', <math>\partial W=M \sqcup N</math>.
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than [[diffeomorphism]] or [[homeomorphism]] of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to [[diffeomorphism]] or [[homeomorphism]] in dimensions ≥ 4 – because the [[word problem for groups]] cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in [[geometric topology]] and [[algebraic topology]]. In geometric topology, cobordisms are [[#Connection with Morse theory|intimately connected]] with [[Morse theory]], and [[h-cobordism|''h''-cobordisms]] are fundamental in the study of high-dimensional manifolds, namely [[surgery theory]]. In algebraic topology, cobordism theories are fundamental [[extraordinary cohomology theories]], and [[Cobordism#Categorical aspects|categories of cobordisms]] are the domains of [[topological quantum field theory|topological quantum field theories]].
== Definition ==
===Manifolds===
Roughly speaking, an ''n''-dimensional [[manifold (mathematics)|manifold]] ''M'' is a [[topological space]] [[neighborhood (mathematics)|locally]] (i.e., near each point) [[homeomorphism|homeomorphic]] to an open subset of [[Euclidean space]] <math>\R^n.</math> A [[manifold with boundary]] is similar, except that a point of ''M'' is allowed to have a neighborhood that is homeomorphic to an open subset of the [[Half-space (geometry)|half-space]]
:<math>\{(x_1,\ldots,x_n) \in \R^n \mid x_n \geqslant 0\}.</math>
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of <math>M</math>; the boundary of <math>M</math> is denoted by <math>\partial M</math>. Finally, a [[closed manifold]] is, by definition, a [[compact space|compact]] manifold without boundary (<math>\partial M=\emptyset</math>.)
===Cobordisms===
An <math>(n+1)</math>-dimensional ''cobordism'' is a [[quintuple]] <math>(W; M, N, i, j)</math> consisting of an <math>(n+1)</math>-dimensional compact differentiable manifold with boundary, <math>W</math>; closed <math>n</math>-manifolds <math>M</math>, <math>N</math>; and [[embedding]]s <math>i\colon M \hookrightarrow \partial W</math>, <math>j\colon N \hookrightarrow\partial W</math> with disjoint images such that
:<math>\partial W = i(M) \sqcup j(N)~.</math>
The terminology is usually abbreviated to <math>(W; M, N)</math>.<ref>The notation "<math>(n+1)</math>-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds.</ref> ''M'' and ''N'' are called ''cobordant'' if such a cobordism exists. All manifolds cobordant to a fixed given manifold ''M'' form the ''cobordism class'' of ''M''.
Every closed manifold ''M'' is the boundary of the non-compact manifold ''M'' × [0, 1); for this reason we require ''W'' to be compact in the definition of cobordism. Note however that ''W'' is ''not'' required to be connected; as a consequence, if ''M'' = ∂''W''<sub>1</sub> and ''N'' = ∂''W''<sub>2</sub>, then ''M'' and ''N'' are cobordant.
===Examples===
The simplest example of a cobordism is the [[unit interval]] {{nowrap|''I'' {{=}} [0, 1]}}. It is a 1-dimensional cobordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold ''M'', ({{nowrap|''M'' × ''I''}}; {{nowrap|''M'' × {0} }}, {{nowrap|''M'' × {1} }}) is a cobordism from ''M'' × {0} to ''M'' × {1}.
[[File:Pair of pants cobordism (pantslike).svg|thumb|right| A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).]]
If ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a [[pair of pants (mathematics)|pair of pants]] ''W'' (see the figure at right). Thus the pair of pants is a cobordism between ''M'' and ''N''. A simpler cobordism between ''M'' and ''N'' is given by the disjoint union of three disks.
The pair of pants is an example of a more general cobordism: for any two ''n''-dimensional manifolds ''M'', ''M''′, the disjoint union <math>M \sqcup M'</math> is cobordant to the [[connected sum]] <math>M\mathbin{\#}M'.</math> The previous example is a particular case, since the connected sum <math>\mathbb{S}^1\mathbin{\#}\mathbb{S}^1</math> is isomorphic to <math>\mathbb{S}^1.</math> The connected sum <math>M\mathbin{\#}M'</math> is obtained from the disjoint union <math>M \sqcup M'</math> by surgery on an embedding of <math>\mathbb{S}^0 \times \mathbb{D}^n</math> in <math>M \sqcup M'</math>, and the cobordism is the trace of the surgery.
===Terminology===
An ''n''-manifold ''M'' is called ''null-cobordant'' if there is a cobordism between ''M'' and the empty manifold; in other words, if ''M'' is the entire boundary of some (''n'' + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a ''n''-sphere is null-cobordant since it bounds a (''n'' + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a [[handlebody]]. On the other hand, the 2''n''-dimensional [[real projective space]] <math>\mathbb{P}^{2n}(\R)</math> is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
The general ''bordism problem'' is to calculate the cobordism classes of manifolds subject to various conditions.
Null-cobordisms with additional structure are called [[symplectic filling|fillings]]. ''Bordism'' and ''cobordism'' are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question ''bordism of manifolds'', and the study of cobordisms as objects ''cobordisms of manifolds''.{{Citation needed|date=March 2012}}
The term ''bordism'' comes from French {{lang|fr|[[wikt:bord|bord]]}}, meaning boundary. Hence bordism is the study of boundaries. ''Cobordism'' means "jointly bound", so ''M'' and ''N'' are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, cobordism groups form an extraordinary ''cohomology theory'', hence the co-.
===Variants===
The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are [[orientability|oriented]], or carry some other additional structure referred to as [[G-structure]]. This gives rise to [[#Oriented cobordism|"oriented cobordism"]] and "cobordism with G-structure", respectively. Under favourable technical conditions these form a [[graded ring]] called the '''cobordism ring''' <math>\Omega^G_*</math>, with grading by dimension, addition by disjoint union and multiplication by [[cartesian product]]. The cobordism groups <math>\Omega^G_*</math> are the coefficient groups of a [[#Cobordism_as_an_extraordinary_cohomology_theory|generalised homology theory]].
When there is additional structure, the notion of cobordism must be formulated more precisely: a ''G''-structure on ''W'' restricts to a ''G''-structure on ''M'' and ''N''. The basic examples are ''G'' = O for unoriented cobordism, ''G'' = SO for oriented cobordism, and ''G'' = U for [[complex cobordism]] using ''stably'' [[complex manifold]]s. Many more are detailed by [[Robert Evert Stong|Robert E. Stong]].<ref>{{Cite book | publisher = [[Princeton University Press]] | last = Stong | first = Robert E. | authorlink=Robert Evert Stong|title=Notes on cobordism theory |location=Princeton, NJ| year = 1968 }}</ref>
In a similar vein, a standard tool in [[surgery theory]] is surgery on [[normal invariants|normal maps]]: such a process changes a normal map to another normal map within the same [[bordism]] class.
Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially [[Piecewise linear manifold|piecewise linear (PL)]] and [[topological manifold]]s. This gives rise to bordism groups <math>\Omega_*^{PL}(X), \Omega_*^{TOP}(X)</math>, which are harder to compute than the differentiable variants.{{citation needed|date=September 2018}}
==Surgery construction==
Recall that in general, if ''X'', ''Y'' are manifolds with boundary, then the boundary of the product manifold is {{nowrap|∂(''X'' × ''Y'') {{=}} (∂''X'' × ''Y'') ∪ (''X'' × ∂''Y'')}}.
Now, given a manifold ''M'' of dimension ''n'' = ''p'' + ''q'' and an [[embedding]] <math>\varphi : \mathbb{S}^p \times \mathbb{D}^q \subset M,</math> define the ''n''-manifold
:<math>N := (M - \operatorname{int~im}\varphi) \cup_{\varphi|_{\mathbb{S}^p\times \mathbb{S}^{q-1}}} \left(\mathbb{D}^{p+1}\times \mathbb{S}^{q-1}\right)</math>
obtained by [[surgery theory|surgery]], via cutting out the interior of <math>\mathbb{S}^p \times \mathbb{D}^q</math> and gluing in <math>\mathbb{D}^{p+1} \times \mathbb{S}^{q-1}</math> along their boundary
:<math>\partial \left (\mathbb{S}^p \times \mathbb{D}^q \right) = \mathbb{S}^p \times \mathbb{S}^{q-1} = \partial \left( \mathbb{D}^{p+1} \times \mathbb{S}^{q-1} \right).</math>
The '''trace''' of the surgery
:<math>W := (M \times I) \cup_{\mathbb{S}^p\times \mathbb{D}^q\times \{1\}} \left(\mathbb{D}^{p+1} \times \mathbb{D}^q\right)</math>
defines an '''elementary''' cobordism (''W''; ''M'', ''N''). Note that ''M'' is obtained from ''N'' by surgery on <math>\mathbb{D}^{p+1}\times \mathbb{S}^{q-1} \subset N.</math> This is called '''reversing the surgery'''.
Every cobordism is a union of elementary cobordisms, by the work of [[Marston Morse]], [[René Thom]] and [[John Milnor]].
===Examples===
[[File:Circle-surgery.svg|thumb|right|Fig. 1]]
As per the above definition, a surgery on the circle consists of cutting out a copy of <math>\mathbb{S}^0 \times \mathbb{D}^1</math> and gluing in <math>\mathbb{D}^1 \times \mathbb{S}^0.</math> The pictures in Fig. 1 show that the result of doing this is either (i) <math>\mathbb{S}^1</math> again, or (ii) two copies of <math>\mathbb{S}^1</math>
[[File:Sphere-surgery1.png|thumb|left|Fig. 2a]]
[[File:Sphere-surgery2.png|thumb|right|Fig. 2b]]
For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either <math>\mathbb{S}^0 \times \mathbb{D}^2</math> or <math>\mathbb{S}^1 \times \mathbb{D}^1.</math>
{{ordered list
| list-style-type = lower-alpha|<math>\mathbb{S}^1 \times \mathbb{D}^1</math>: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in <math>\mathbb{S}^0 \times \mathbb{D}^2</math> – that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)|[[File:Sphere-surgery4.png|thumb|right|Fig. 2c. This shape cannot be embedded in 3-space.]]
<math>\mathbb{S}^0 \times \mathbb{D}^2</math>: Having cut out two disks <math>\mathbb{S}^0 \times \mathbb{D}^2,</math> we glue back in the cylinder <math>\mathbb{S}^1 \times \mathbb{D}^1.</math> There are two possible outcomes, depending on whether our gluing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the [[torus]] <math>\mathbb{S}^1 \times \mathbb{S}^1</math> but if they are different, we obtain the [[Klein bottle]] (Fig. 2c).
}}
==Morse functions==
Suppose that ''f'' is a [[Morse function]] on an (''n'' + 1)-dimensional manifold, and suppose that ''c'' is a critical value with exactly one critical point in its preimage. If the index of this critical point is ''p'' + 1, then the level-set ''N'' := ''f''<sup>−1</sup>(''c'' + ε) is obtained from ''M'' := ''f''<sup>−1</sup>(''c'' − ε) by a ''p''-surgery. The inverse image ''W'' := ''f''<sup>−1</sup>([''c'' − ε, ''c'' + ε]) defines a cobordism (''W''; ''M'', ''N'') that can be identified with the trace of this surgery.
===Geometry, and the connection with Morse theory and handlebodies===
Given a cobordism (''W''; ''M'', ''N'') there exists a smooth function ''f'' : ''W'' → [0, 1] such that ''f''<sup>−1</sup>(0) = ''M'', ''f''<sup>−1</sup>(1) = ''N''. By general position, one can assume ''f'' is Morse and such that all critical points occur in the interior of ''W''. In this setting ''f'' is called a Morse function on a cobordism. The cobordism (''W''; ''M'', ''N'') is a union of the traces of a sequence of surgeries on ''M'', one for each critical point of ''f''. The manifold ''W'' is obtained from ''M'' × [0, 1] by attaching one [[handle decomposition|handle]] for each critical point of ''f''.
[[File:Cobordism.svg|thumb|The 3-dimensional cobordism <math>W = \mathbb{S}^1 \times \mathbb{D}^2 - \mathbb{D}^3</math> between the 2-[[sphere]] <math>M = \mathbb{S}^2</math> and the 2-[[torus]] <math>N = \mathbb{S}^1 \times \mathbb{S}^1,</math> with ''N'' obtained from ''M'' by surgery on <math>\mathbb{S}^0 \times \mathbb{D}^2 \subset M,</math>and ''W'' obtained from ''M'' × ''I'' by attaching a 1-handle <math>\mathbb{D}^1 \times \mathbb{D}^2.</math>]]
The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of ''f''′ give rise to a [[handle decomposition|handle presentation]] of the triple (''W''; ''M'', ''N''). Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.
==History==
Cobordism had its roots in the (failed) attempt by [[Henri Poincaré]] in 1895 to define [[homology (mathematics)|homology]] purely in terms of manifolds {{harv|Dieudonné|1989|loc=[https://archive.org/details/historyofalgebra0000dieu_g9a3/page/290 p. 289]}}. Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See [[#Cobordism as an extraordinary cohomology theory|Cobordism as an extraordinary cohomology theory]] for the relationship between bordism and homology.
Bordism was explicitly introduced by [[Lev Pontryagin]] in geometric work on manifolds. It came to prominence when [[René Thom]] showed that cobordism groups could be computed by means of [[homotopy theory]], via the [[Thom complex]] construction. Cobordism theory became part of the apparatus of [[extraordinary cohomology theory]], alongside [[K-theory]]. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the [[Hirzebruch–Riemann–Roch theorem]], and in the first proofs of the [[Atiyah–Singer index theorem]].
In the 1980s the [[category (mathematics)|category]] with compact manifolds as [[object (category theory)|objects]] and cobordisms between these as [[morphism]]s played a basic role in the Atiyah–Segal axioms for [[topological quantum field theory]], which is an important part of [[quantum topology]].
==Categorical aspects==
Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a [[category (mathematics)|category]] whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of (''W''; ''M'', ''N'') and (''W'' ′; ''N'', ''P'') is defined by gluing the right end of the first to the left end of the second, yielding (''W'' ′ ∪<sub>''N''</sub> ''W''; ''M'', ''P''). A cobordism is a kind of [[cospan]]:<ref>While every cobordism is a cospan, the category of cobordisms is ''not'' a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that ''M'' and ''N'' form a partition of the boundary of ''W'' is a global constraint.</ref> ''M'' → ''W'' ← ''N''. The category is a [[dagger compact category]].
A [[topological quantum field theory]] is a [[monoidal functor]] from a category of cobordisms to a category of [[vector space]]s. That is, it is a [[functor]] whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a nullary (0-ary) operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
==Unoriented cobordism==
{{Further|List of cohomology theories#Unoriented cobordism}}
The set of cobordism classes of closed unoriented ''n''-dimensional manifolds is usually denoted by <math>\mathfrak{N}_n</math> (rather than the more systematic <math>\Omega_n^{\text{O}}</math>); it is an [[abelian group]] with the disjoint union as operation. More specifically, if [''M''] and [''N''] denote the cobordism classes of the manifolds ''M'' and ''N'' respectively, we define <math>[M]+[N] = [M \sqcup N]</math>; this is a well-defined operation which turns <math>\mathfrak{N}_n</math> into an abelian group. The identity element of this group is the class <math>[\emptyset]</math> consisting of all closed ''n''-manifolds which are boundaries. Further we have <math>[M] + [M] = [\emptyset]</math> for every ''M'' since <math>M \sqcup M = \partial (M \times [0,1])</math>. Therefore, <math>\mathfrak{N}_n</math> is a vector space over <math>\mathbb{F}_2</math>, the [[GF(2)|field with two elements]]. The cartesian product of manifolds defines a multiplication <math>[M][N]=[M \times N],</math> so
:<math>\mathfrak{N}_* = \bigoplus_{n \geqslant 0}\mathfrak{N}_n</math>
is a [[graded algebra]], with the grading given by the dimension.
The cobordism class <math>[M] \in \mathfrak{N}_n</math> of a closed unoriented ''n''-dimensional manifold ''M'' is determined by the Stiefel–Whitney [[characteristic number]]s of ''M'', which depend on the stable isomorphism class of the [[tangent bundle]]. Thus if ''M'' has a stably trivial tangent bundle then <math>[M]=0 \in \mathfrak{N}_n</math>. In 1954 [[René Thom]] proved
:<math>\mathfrak{N}_* = \mathbb{F}_2 \left[x_i | i \geqslant 1, i \neq 2^j - 1 \right]</math>
the polynomial algebra with one generator <math>x_i</math> in each dimension <math>i \neq 2^j - 1</math>. Thus two unoriented closed ''n''-dimensional manifolds ''M'', ''N'' are cobordant, <math>[M] = [N] \in \mathfrak{N}_n,</math> if and only if for each collection <math>\left(i_1, \cdots, i_k\right)</math> of ''k''-tuples of integers <math>i \geqslant 1, i \neq 2^j - 1</math> such that <math>i_1 + \cdots + i_k = n</math> the Stiefel-Whitney numbers are equal
:<math>\left\langle w_{i_1}(M) \cdots w_{i_k}(M), [M] \right\rangle = \left\langle w_{i_1}(N) \cdots w_{i_k}(N), [N] \right\rangle \in \mathbb{F}_2</math>
with <math>w_i(M) \in H^i\left(M; \mathbb{F}_2\right)</math> the ''i''th [[Stiefel-Whitney class]] and <math>[M] \in H_n\left(M; \mathbb{F}_2\right)</math> the <math>\mathbb{F}_2</math>-coefficient [[fundamental class]].
For even ''i'' it is possible to choose <math>x_i = \left[\mathbb{P}^i(\R)\right]</math>, the cobordism class of the ''i''-dimensional [[real projective space]].
The low-dimensional unoriented cobordism groups are
:<math>\begin{align}
\mathfrak{N}_0 &= \Z/2, \\
\mathfrak{N}_1 &= 0, \\
\mathfrak{N}_2 &= \Z/2, \\
\mathfrak{N}_3 &= 0, \\
\mathfrak{N}_4 &= \Z/2 \oplus \Z/2, \\
\mathfrak{N}_5 &= \Z/2.
\end{align}</math>
This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).
The [[Euler characteristic]] <math>\chi(M) \in \Z</math> modulo 2 of an unoriented manifold ''M'' is an unoriented cobordism invariant. This is implied by the equation
:<math>\chi_{\partial W} = \left(1 - (-1)^{\dim W} \right)\chi_W</math>
for any compact manifold with boundary <math>W</math>.
Therefore, <math>\chi: \mathfrak{N}_i \to \Z/2</math> is a well-defined group homomorphism. For example, for any <math>i_1, \cdots, i_k \in\mathbb{N}</math>
:<math>\chi \left( \mathbb{P}^{2i_1} (\R) \times \cdots \times \mathbb{P}^{2i_k}(\R) \right) = 1.</math>
In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map <math>\chi: \mathfrak{N}_{2i} \to \Z/2</math> is onto for all <math>i \in \mathbb{N},</math> and a group isomorphism for <math>i = 1.</math>
Moreover, because of <math>\chi(M \times N) = \chi(M)\chi(N)</math>, these group homomorphism assemble into a homomorphism of graded algebras:
:<math>\begin{cases}
\mathfrak{N} \to \mathbb{F}_2[x] \\[]
[M] \mapsto \chi(M) x^{\dim(M)}
\end{cases}</math>
==Cobordism of manifolds with additional structure==
Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of ''X''-structure (or [[G-structure]]).<ref>{{Citation | last1=Switzer | first1=Robert M. | title=Algebraic topology—homotopy and homology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 | mr=1886843 | year=2002}}, chapter 12</ref> Very briefly, the [[normal bundle]] ν of an immersion of ''M'' into a sufficiently high-dimensional [[Euclidean space]] <math>\R^{n+k}</math> gives rise to a map from ''M'' to the [[Grassmannian]], which in turn is a subspace of the [[classifying space]] of the [[orthogonal group]]: ν: ''M'' → '''Gr'''(''n'', ''n'' + ''k'') → ''BO''(''k''). Given a collection of spaces and maps ''X<sub>k</sub>'' → ''X<sub>k</sub>''<sub>+1</sub> with maps ''X<sub>k</sub>'' → ''BO''(''k'') (compatible with the inclusions ''BO''(''k'') → ''BO''(''k''+1), an ''X''-structure is a lift of ν to a map <math>\tilde \nu: M \to X_k</math>. Considering only manifolds and cobordisms with ''X''-structure gives rise to a more general notion of cobordism. In particular, ''X<sub>k</sub>'' may be given by ''BG''(''k''), where ''G''(''k'') → ''O''(''k'') is some group homomorphism. This is referred to as a [[G-structure]]. Examples include ''G'' = ''O'', the orthogonal group, giving back the unoriented cobordism, but also the subgroup [[special linear group|SO(''k'')]], giving rise to [[oriented cobordism]], the [[spin group]], the [[unitary group|unitary group ''U''(''k'')]], and the trivial group, giving rise to [[framed cobordism]].
The resulting cobordism groups are then defined analogously to the unoriented case. They are denoted by <math>\Omega^G_*</math>.
===Oriented cobordism===
{{Further|List of cohomology theories#Oriented cobordism}}
Oriented cobordism is the one of manifolds with an SO-structure. Equivalently, all manifolds need to be [[orientability|oriented]] and cobordisms (''W'', ''M'', ''N'') (also referred to as ''oriented cobordisms'' for clarity) are such that the boundary (with the induced orientations) is <math>M \sqcup (-N)</math>, where −''N'' denotes ''N'' with the reversed orientation. For example, boundary of the cylinder ''M'' × ''I'' is <math>M \sqcup (-M)</math>: both ends have opposite orientations. It is also the correct definition in the sense of [[extraordinary cohomology theory]].
Unlike in the unoriented cobordism group, where every element is two-torsion, 2''M'' is not in general an oriented boundary, that is, 2[''M''] ≠ 0 when considered in <math>\Omega_*^{\text{SO}}.</math>
The oriented cobordism groups are given modulo torsion by
:<math>\Omega_*^{\text{SO}}\otimes \Q =\Q \left [y_{4i}\mid i \geqslant 1 \right ],</math>
the polynomial algebra generated by the oriented cobordism classes
:<math>y_{4i}=\left [\mathbb{P}^{2i}(\Complex) \right ] \in \Omega_{4i}^{\text{SO}}</math>
of the [[complex projective space]]s (Thom, 1952). The oriented cobordism group <math>\Omega_*^{\text{SO}}</math> is determined by the Stiefel–Whitney and Pontrjagin [[characteristic number]]s (Wall, 1960). Two oriented manifolds are oriented cobordant if and only if their Stiefel–Whitney and Pontrjagin numbers are the same.
The low-dimensional oriented cobordism groups are :
:<math>\begin{align}
\Omega_0^{\text{SO}} &= \Z, \\
\Omega_1^{\text{SO}} &= 0, \\
\Omega_2^{\text{SO}} &= 0, \\
\Omega_3^{\text{SO}} &= 0, \\
\Omega_4^{\text{SO}} &= \Z, \\
\Omega_5^{\text{SO}} &= \Z_2.
\end{align}</math>
The [[Signature of a manifold|signature]] of an oriented 4''i''-dimensional manifold ''M'' is defined as the signature of the intersection form on <math>H^{2i}(M) \in \Z</math> and is denoted by <math>\sigma(M).</math> It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the [[Hirzebruch signature theorem]].
For example, for any ''i''<sub>1</sub>, ..., ''i<sub>k</sub>'' ≥ 1
:<math>\sigma \left (\mathbb{P}^{2i_1}(\Complex) \times \cdots \times \mathbb{P}^{2i_k}(\Complex) \right) = 1.</math>
The signature map <math>\sigma:\Omega_{4i}^{\text{SO}} \to \Z</math> is onto for all ''i'' ≥ 1, and an isomorphism for ''i'' = 1.
==Cobordism as an extraordinary cohomology theory==
Every [[vector bundle]] theory (real, complex etc.) has an [[extraordinary cohomology theory]] called [[K-theory]]. Similarly, every cobordism theory Ω<sup>''G''</sup> has an [[extraordinary cohomology theory]], with homology ("bordism") groups <math>\Omega^G_n(X)</math> and cohomology ("cobordism") groups <math>\Omega^n_G(X)</math> for any space ''X''. The generalized homology groups <math>\Omega_*^G(X)</math> are [[Covariance|covariant]] in ''X'', and the generalized cohomology groups <math>\Omega^*_G(X)</math> are [[Covariance and contravariance of vectors|contravariant]] in ''X''. The cobordism groups defined above are, from this point of view, the homology groups of a point: <math>\Omega_n^G = \Omega_n^G(\text{pt})</math>. Then <math>\Omega^G_n(X)</math> is the group of ''bordism'' classes of pairs (''M'', ''f'') with ''M'' a closed ''n''-dimensional manifold ''M'' (with G-structure) and ''f'' : ''M'' → ''X'' a map. Such pairs (''M'', ''f''), (''N'', ''g'') are ''bordant'' if there exists a G-cobordism (''W''; ''M'', ''N'') with a map ''h'' : ''W'' → ''X'', which restricts to ''f'' on ''M'', and to ''g'' on ''N''.
An ''n''-dimensional manifold ''M'' has a [[Homology (mathematics)|fundamental homology class]] [''M''] ∈ ''H<sub>n</sub>''(''M'') (with coefficients in <math>\Z/2</math> in general, and in <math>\Z</math> in the oriented case), defining a natural transformation
:<math>\begin{cases}
\Omega^G_n(X) \to H_n(X) \\
(M,f) \mapsto f_*[M]
\end{cases}</math>
which is far from being an isomorphism in general.
The bordism and cobordism theories of a space satisfy the [[Eilenberg–Steenrod axioms]] apart from the dimension axiom. This does not mean that the groups <math>\Omega^n_G(X)</math> can be effectively computed once one knows the cobordism theory of a point and the homology of the space ''X'', though the [[Atiyah–Hirzebruch spectral sequence]] gives a starting point for calculations. The computation is only easy if the particular cobordism theory [[#Cobordism_as_an_extraordinary_cohomology_theory|reduces to a product of ordinary homology theories]], in which case the bordism groups are the ordinary homology groups
:<math>\Omega^G_n(X)=\sum_{p+q=n}H_p(X;\Omega^G_q(\text{pt})).</math>
This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably [[Pontrjagin–Thom construction#Framed cobordism|framed cobordism]], oriented cobordism and [[complex cobordism]]. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the [[homotopy groups of spheres]]).<ref>{{Cite book |first=D.C. |last=Ravenel |title=Complex cobordism and stable homotopy groups of spheres |publisher=Academic Press |date=April 1986 |isbn=0-12-583430-6 }}</ref>
Cobordism theories are represented by [[Thom spectrum|Thom spectra]] ''MG'': given a group ''G'', the Thom spectrum is composed from the [[Thom space]]s ''MG<sub>n</sub>'' of the [[tautological bundle|standard vector bundles]] over the [[classifying space]]s ''BG<sub>n</sub>''. Note that even for similar groups, Thom spectra can be very different: ''MSO'' and ''MO'' are very different, reflecting the difference between oriented and unoriented cobordism.
From the point of view of spectra, unoriented cobordism is a product of [[Eilenberg–MacLane spectrum|Eilenberg–MacLane spectra]] – ''MO'' = ''H''({{pi}}<sub>∗</sub>(''MO'')) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum ''MSO'' is rather more complicated than ''MO''.
==See also==
*[[h-cobordism|''h''-cobordism]]
*[[Link concordance]]
*[[List of cohomology theories]]
*[[Symplectic filling]]
*[[Cobordism hypothesis]]
*[[Cobordism ring]]
*[[Timeline of bordism]]
==Notes==
{{Reflist}}
==References==
*[[Frank Adams|John Frank Adams]], ''Stable homotopy and generalised homology'', Univ. Chicago Press (1974).
*{{springer
| title= bordism
| id=b/b017030
| last= Anosov
| first= Dmitri V.
| author-link= Dmitri Anosov
| last2= Voitsekhovskii
| first2= M. I.
| author2-link=
}}
*[[Michael Atiyah|Michael F. Atiyah]], ''Bordism and cobordism'' Proc. Camb. Phil. Soc. 57, pp. 200–208 (1961).
*{{Cite book
|first=Jean Alexandre
|last=Dieudonné
|authorlink=Jean Dieudonné
|title=A history of algebraic and differential topology, 1900–1960
|isbn=978-0-8176-3388-2
|publisher=Birkhäuser
|location=Boston
|year=1989
|url-access=registration
|url=https://archive.org/details/historyofalgebra0000dieu_g9a3
}}
*{{Cite document
|first=Antoni A.
|last=Kosinski
|title=Differential Manifolds
|publisher=Dover Publications
|date=October 19, 2007
}}
*{{Cite book
|first1=Ib
|last1=Madsen
|author1-link = Ib Madsen
|first2=R. James
|last2=Milgram
|title=The classifying spaces for surgery and cobordism of manifolds
|publisher=[[Princeton University Press]]
|location=[[Princeton, New Jersey]]
|isbn=978-0-691-08226-4
|year=1979
}}
*{{Cite journal
| issn = 0013-8584
| volume = 8
| pages = 16–23
| last = Milnor
| first = John
| authorlink=John Milnor
| title = A survey of cobordism theory
| journal = [[L'Enseignement Mathématique]]
| year = 1962
}}
*[[Sergei Novikov (mathematician)|Sergei Novikov]], ''Methods of algebraic topology from the point of view of cobordism theory'', Izv. Akad. Nauk SSSR Ser. Mat. '''31''' (1967), 855–951.
*[[Lev Pontryagin]], ''Smooth manifolds and their applications in homotopy theory'' American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959).
*[[Daniel Quillen]], ''On the formal group laws of unoriented and complex cobordism theory'' Bull. Amer. Math. Soc., 75 (1969) pp. 1293–1298.
*[[Douglas Ravenel]], ''Complex cobordism and stable homotopy groups of spheres'', Acad. Press (1986).
*{{springer|id=C/c022780|title=Cobordism|author=Yuli B. Rudyak}}
*[[Yuli Rudyak|Yuli B. Rudyak]], ''On Thom spectra, orientability, and (co)bordism'', Springer (2008).
* [[Robert Evert Stong|Robert E. Stong]], ''Notes on cobordism theory'', Princeton Univ. Press (1968).
*{{Cite book
| publisher = World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ
| isbn = 978-981-270-559-4
| volume = 39
| last = Taimanov
| first = Iskander A.
| others = [[Sergei Novikov (mathematician)|S. Novikov]] (ed.)
| title = Topological library. Part 1: cobordisms and their applications
| series = Series on Knots and Everything
| year = 2007
}}
* [[René Thom]], ''Quelques propriétés globales des variétés différentiables'', [[Commentarii Mathematici Helvetici]] 28, 17-86 (1954).
*{{Cite journal
| issn = 0003-486X
| volume = 72
| pages = 292–311
| last =Wall
| first =C. T. C.
| authorlink=C. T. C. Wall
| title = Determination of the cobordism ring
| journal = [[Annals of Mathematics]] |series=Second Series
| year = 1960
| doi = 10.2307/1970136
| issue = 2
| publisher = The Annals of Mathematics, Vol. 72, No. 2
| jstor = 1970136
}}
== External links ==
* [https://web.archive.org/web/20110719102848/http://www.map.him.uni-bonn.de/Bordism Bordism] on the Manifold Atlas.
* [https://archive.today/20120529222452/http://www.map.him.uni-bonn.de/B-Bordism B-Bordism] on the Manifold Atlas.
{{Topology}}
{{Authority control}}
[[Category:Differential topology]]
[[Category:Algebraic topology]]
[[Category:Surgery theory]]' |
New page wikitext, after the edit (new_wikitext) | '[[File:Cobordism.svg|thumb|A cobordism (''W''; ''M'', ''N'').]]
In [[mathematics]], '''cobordism''' is a fundamental [[equivalence relation]] on the class of [[compact space|compact]] [[manifold]]s of the same dimension, set up using the concept of the [[boundary (topology)|boundary]] (French ''[[wikt:bord#French|bord]]'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their [[disjoint union]] is the ''boundary'' of a compact manifold one dimension higher.
The boundary of an (''n'' + 1)-dimensional [[manifold]] ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by [[René Thom]] for [[smooth manifold]]s (i.e., differentiable), but there are now also versions for
[[Piecewise linear manifold|piecewise linear]] and [[topological manifold]]s.
A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', <math>\partial W=M \sqcup N</math>.
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than [[diffeomorphism]] or [[homeomorphism]] of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to [[diffeomorphism]] or [[homeomorphism]] in dimensions ≥ 4 – because the [[word problem for groups]] cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in [[geometric topology]] and [[algebraic topology]]. In geometric topology, cobordisms are [[#Connection with Morse theory|intimately connected]] with [[Morse theory]], and [[h-cobordism|''h''-cobordisms]] are fundamental in the study of high-dimensional manifolds, namely [[surgery theory]]. In algebraic topology, cobordism theories are fundamental [[extraordinary cohomology theories]], and [[Cobordism#Categorical aspects|categories of cobordisms]] are the domains of [[topological quantum field theory|topological quantum field theories]].
== Definition ==
===Manifolds===
Roughly speaking, an ''n''-dimensional [[manifold (mathematics)|manifold]] ''M'' is a [[topological space]] [[neighborhood (mathematics)|locally]] (i.e., near each point) [[homeomorphism|homeomorphic]] to an open subset of [[Euclidean space]] <math>\R^n.</math> A [[manifold with boundary]] is similar, except that a point of ''M'' is allowed to have a neighborhood that is homeomorphic to an open subset of the [[Half-space (geometry)|half-space]]
:<math>\{(x_1,\ldots,x_n) \in \R^n \mid x_n \geqslant 0\}.</math>
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of <math>M</math>; the boundary of <math>M</math> is denoted by <math>\partial M</math>. Finally, a [[closed manifold]] is, by definition, a [[compact space|compact]] manifold without boundary (<math>\partial M=\emptyset</math>.)
===Bordisms===
An <math>(n+1)</math>-dimensional ''bordism'' is a [[quintuple]] <math>(W; M, N, i, j)</math> consisting of an <math>(n+1)</math>-dimensional compact differentiable manifold with boundary, <math>W</math>; closed <math>n</math>-manifolds <math>M</math>, <math>N</math>; and [[embedding]]s <math>i\colon M \hookrightarrow \partial W</math>, <math>j\colon N \hookrightarrow\partial W</math> with disjoint images such that
:<math>\partial W = i(M) \sqcup j(N)~.</math>
The terminology is usually abbreviated to <math>(W; M, N)</math>.<ref>The notation "<math>(n+1)</math>-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional bordism" refers to a 5-dimensional bordism between 4-dimensional manifolds or a 6-dimensional bordism between 5-dimensional manifolds.</ref> ''M'' and ''N'' are called ''cobordant'' if such a cobordism exists. All manifolds cobordant to a fixed given manifold ''M'' form the ''bordism class'' of ''M''.
Every closed manifold ''M'' is the boundary of the non-compact manifold ''M'' × [0, 1); for this reason we require ''W'' to be compact in the definition of bordism. Note however that ''W'' is ''not'' required to be connected; as a consequence, if ''M'' = ∂''W''<sub>1</sub> and ''N'' = ∂''W''<sub>2</sub>, then ''M'' and ''N'' are cobordant.
===Examples===
The simplest example of a cobordism is the [[unit interval]] {{nowrap|''I'' {{=}} [0, 1]}}. It is a 1-dimensional bordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold ''M'', ({{nowrap|''M'' × ''I''}}; {{nowrap|''M'' × {0} }}, {{nowrap|''M'' × {1} }}) is a cobordism from ''M'' × {0} to ''M'' × {1}.
[[File:Pair of pants bordism (pantslike).svg|thumb|right| A bordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).]]
If ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a [[pair of pants (mathematics)|pair of pants]] ''W'' (see the figure at right). Thus the pair of pants is a cobordism between ''M'' and ''N''. A simpler bordism between ''M'' and ''N'' is given by the disjoint union of three disks.
The pair of pants is an example of a more general bordism: for any two ''n''-dimensional manifolds ''M'', ''M''′, the disjoint union <math>M \sqcup M'</math> is cobordant to the [[connected sum]] <math>M\mathbin{\#}M'.</math> The previous example is a particular case, since the connected sum <math>\mathbb{S}^1\mathbin{\#}\mathbb{S}^1</math> is isomorphic to <math>\mathbb{S}^1.</math> The connected sum <math>M\mathbin{\#}M'</math> is obtained from the disjoint union <math>M \sqcup M'</math> by surgery on an embedding of <math>\mathbb{S}^0 \times \mathbb{D}^n</math> in <math>M \sqcup M'</math>, and the bordism is the trace of the surgery.
===Terminology===
An ''n''-manifold ''M'' is called ''null-cobordant'' if there is a bordism between ''M'' and the empty manifold; in other words, if ''M'' is the entire boundary of some (''n'' + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a ''n''-sphere is null-cobordant since it bounds a (''n'' + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a [[handlebody]]. On the other hand, the 2''n''-dimensional [[real projective space]] <math>\mathbb{P}^{2n}(\R)</math> is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
The general ''bordism problem'' is to calculate the cobordism classes of manifolds subject to various conditions.
Null-bordisms with additional structure are called [[symplectic filling|fillings]]. ''Bordism'' and ''Cobordism'' are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question ''bordism of manifolds'', and the study of cobordisms as objects ''cobordisms of manifolds''.{{Citation needed|date=March 2012}}
The term ''bordism'' comes from French {{lang|fr|[[wikt:bord|bord]]}}, meaning boundary. Hence bordism is the study of boundaries. ''Cobordism'' means "jointly bound", so ''M'' and ''N'' are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, bordism groups form an extraordinary ''cohomology theory'', hence the co-.
===Variants===
The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are [[orientability|oriented]], or carry some other additional structure referred to as [[G-structure]]. This gives rise to [[#Oriented bordism|"oriented bordism"]] and "bordism with G-structure", respectively. Under favourable technical conditions these form a [[graded ring]] called the '''bordism ring''' <math>\Omega^G_*</math>, with grading by dimension, addition by disjoint union and multiplication by [[cartesian product]]. The bordism groups <math>\Omega^G_*</math> are the coefficient groups of a [[#Cobordism_as_an_extraordinary_cohomology_theory|generalised homology theory]].
When there is additional structure, the notion of bordism must be formulated more precisely: a ''G''-structure on ''W'' restricts to a ''G''-structure on ''M'' and ''N''. The basic examples are ''G'' = O for unoriented bordism, ''G'' = SO for oriented bordism, and ''G'' = U for [[complex bordism]] using ''stably'' [[complex manifold]]s. Many more are detailed by [[Robert Evert Stong|Robert E. Stong]].<ref>{{Cite book | publisher = [[Princeton University Press]] | last = Stong | first = Robert E. | authorlink=Robert Evert Stong|title=Notes on bordism theory |location=Princeton, NJ| year = 1968 }}</ref>
In a similar vein, a standard tool in [[surgery theory]] is surgery on [[normal invariants|normal maps]]: such a process changes a normal map to another normal map within the same [[bordism]] class.
Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially [[Piecewise linear manifold|piecewise linear (PL)]] and [[topological manifold]]s. This gives rise to bordism groups <math>\Omega_*^{PL}(X), \Omega_*^{TOP}(X)</math>, which are harder to compute than the differentiable variants.{{citation needed|date=September 2018}}
==Surgery construction==
Recall that in general, if ''X'', ''Y'' are manifolds with boundary, then the boundary of the product manifold is {{nowrap|∂(''X'' × ''Y'') {{=}} (∂''X'' × ''Y'') ∪ (''X'' × ∂''Y'')}}.
Now, given a manifold ''M'' of dimension ''n'' = ''p'' + ''q'' and an [[embedding]] <math>\varphi : \mathbb{S}^p \times \mathbb{D}^q \subset M,</math> define the ''n''-manifold
:<math>N := (M - \operatorname{int~im}\varphi) \cup_{\varphi|_{\mathbb{S}^p\times \mathbb{S}^{q-1}}} \left(\mathbb{D}^{p+1}\times \mathbb{S}^{q-1}\right)</math>
obtained by [[surgery theory|surgery]], via cutting out the interior of <math>\mathbb{S}^p \times \mathbb{D}^q</math> and gluing in <math>\mathbb{D}^{p+1} \times \mathbb{S}^{q-1}</math> along their boundary
:<math>\partial \left (\mathbb{S}^p \times \mathbb{D}^q \right) = \mathbb{S}^p \times \mathbb{S}^{q-1} = \partial \left( \mathbb{D}^{p+1} \times \mathbb{S}^{q-1} \right).</math>
The '''trace''' of the surgery
:<math>W := (M \times I) \cup_{\mathbb{S}^p\times \mathbb{D}^q\times \{1\}} \left(\mathbb{D}^{p+1} \times \mathbb{D}^q\right)</math>
defines an '''elementary''' cobordism (''W''; ''M'', ''N''). Note that ''M'' is obtained from ''N'' by surgery on <math>\mathbb{D}^{p+1}\times \mathbb{S}^{q-1} \subset N.</math> This is called '''reversing the surgery'''.
Every cobordism is a union of elementary cobordisms, by the work of [[Marston Morse]], [[René Thom]] and [[John Milnor]].
===Examples===
[[File:Circle-surgery.svg|thumb|right|Fig. 1]]
As per the above definition, a surgery on the circle consists of cutting out a copy of <math>\mathbb{S}^0 \times \mathbb{D}^1</math> and gluing in <math>\mathbb{D}^1 \times \mathbb{S}^0.</math> The pictures in Fig. 1 show that the result of doing this is either (i) <math>\mathbb{S}^1</math> again, or (ii) two copies of <math>\mathbb{S}^1</math>
[[File:Sphere-surgery1.png|thumb|left|Fig. 2a]]
[[File:Sphere-surgery2.png|thumb|right|Fig. 2b]]
For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either <math>\mathbb{S}^0 \times \mathbb{D}^2</math> or <math>\mathbb{S}^1 \times \mathbb{D}^1.</math>
{{ordered list
| list-style-type = lower-alpha|<math>\mathbb{S}^1 \times \mathbb{D}^1</math>: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in <math>\mathbb{S}^0 \times \mathbb{D}^2</math> – that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)|[[File:Sphere-surgery4.png|thumb|right|Fig. 2c. This shape cannot be embedded in 3-space.]]
<math>\mathbb{S}^0 \times \mathbb{D}^2</math>: Having cut out two disks <math>\mathbb{S}^0 \times \mathbb{D}^2,</math> we glue back in the cylinder <math>\mathbb{S}^1 \times \mathbb{D}^1.</math> There are two possible outcomes, depending on whether our gluing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the [[torus]] <math>\mathbb{S}^1 \times \mathbb{S}^1</math> but if they are different, we obtain the [[Klein bottle]] (Fig. 2c).
}}
==Morse functions==
Suppose that ''f'' is a [[Morse function]] on an (''n'' + 1)-dimensional manifold, and suppose that ''c'' is a critical value with exactly one critical point in its preimage. If the index of this critical point is ''p'' + 1, then the level-set ''N'' := ''f''<sup>−1</sup>(''c'' + ε) is obtained from ''M'' := ''f''<sup>−1</sup>(''c'' − ε) by a ''p''-surgery. The inverse image ''W'' := ''f''<sup>−1</sup>([''c'' − ε, ''c'' + ε]) defines a cobordism (''W''; ''M'', ''N'') that can be identified with the trace of this surgery.
===Geometry, and the connection with Morse theory and handlebodies===
Given a cobordism (''W''; ''M'', ''N'') there exists a smooth function ''f'' : ''W'' → [0, 1] such that ''f''<sup>−1</sup>(0) = ''M'', ''f''<sup>−1</sup>(1) = ''N''. By general position, one can assume ''f'' is Morse and such that all critical points occur in the interior of ''W''. In this setting ''f'' is called a Morse function on a cobordism. The cobordism (''W''; ''M'', ''N'') is a union of the traces of a sequence of surgeries on ''M'', one for each critical point of ''f''. The manifold ''W'' is obtained from ''M'' × [0, 1] by attaching one [[handle decomposition|handle]] for each critical point of ''f''.
[[File:Cobordism.svg|thumb|The 3-dimensional cobordism <math>W = \mathbb{S}^1 \times \mathbb{D}^2 - \mathbb{D}^3</math> between the 2-[[sphere]] <math>M = \mathbb{S}^2</math> and the 2-[[torus]] <math>N = \mathbb{S}^1 \times \mathbb{S}^1,</math> with ''N'' obtained from ''M'' by surgery on <math>\mathbb{S}^0 \times \mathbb{D}^2 \subset M,</math>and ''W'' obtained from ''M'' × ''I'' by attaching a 1-handle <math>\mathbb{D}^1 \times \mathbb{D}^2.</math>]]
The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of ''f''′ give rise to a [[handle decomposition|handle presentation]] of the triple (''W''; ''M'', ''N''). Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.
==History==
Cobordism had its roots in the (failed) attempt by [[Henri Poincaré]] in 1895 to define [[homology (mathematics)|homology]] purely in terms of manifolds {{harv|Dieudonné|1989|loc=[https://archive.org/details/historyofalgebra0000dieu_g9a3/page/290 p. 289]}}. Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See [[#Cobordism as an extraordinary cohomology theory|Cobordism as an extraordinary cohomology theory]] for the relationship between bordism and homology.
Bordism was explicitly introduced by [[Lev Pontryagin]] in geometric work on manifolds. It came to prominence when [[René Thom]] showed that cobordism groups could be computed by means of [[homotopy theory]], via the [[Thom complex]] construction. Cobordism theory became part of the apparatus of [[extraordinary cohomology theory]], alongside [[K-theory]]. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the [[Hirzebruch–Riemann–Roch theorem]], and in the first proofs of the [[Atiyah–Singer index theorem]].
In the 1980s the [[category (mathematics)|category]] with compact manifolds as [[object (category theory)|objects]] and cobordisms between these as [[morphism]]s played a basic role in the Atiyah–Segal axioms for [[topological quantum field theory]], which is an important part of [[quantum topology]].
==Categorical aspects==
Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a [[category (mathematics)|category]] whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of (''W''; ''M'', ''N'') and (''W'' ′; ''N'', ''P'') is defined by gluing the right end of the first to the left end of the second, yielding (''W'' ′ ∪<sub>''N''</sub> ''W''; ''M'', ''P''). A cobordism is a kind of [[cospan]]:<ref>While every cobordism is a cospan, the category of cobordisms is ''not'' a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that ''M'' and ''N'' form a partition of the boundary of ''W'' is a global constraint.</ref> ''M'' → ''W'' ← ''N''. The category is a [[dagger compact category]].
A [[topological quantum field theory]] is a [[monoidal functor]] from a category of cobordisms to a category of [[vector space]]s. That is, it is a [[functor]] whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a nullary (0-ary) operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
==Unoriented cobordism==
{{Further|List of cohomology theories#Unoriented cobordism}}
The set of cobordism classes of closed unoriented ''n''-dimensional manifolds is usually denoted by <math>\mathfrak{N}_n</math> (rather than the more systematic <math>\Omega_n^{\text{O}}</math>); it is an [[abelian group]] with the disjoint union as operation. More specifically, if [''M''] and [''N''] denote the cobordism classes of the manifolds ''M'' and ''N'' respectively, we define <math>[M]+[N] = [M \sqcup N]</math>; this is a well-defined operation which turns <math>\mathfrak{N}_n</math> into an abelian group. The identity element of this group is the class <math>[\emptyset]</math> consisting of all closed ''n''-manifolds which are boundaries. Further we have <math>[M] + [M] = [\emptyset]</math> for every ''M'' since <math>M \sqcup M = \partial (M \times [0,1])</math>. Therefore, <math>\mathfrak{N}_n</math> is a vector space over <math>\mathbb{F}_2</math>, the [[GF(2)|field with two elements]]. The cartesian product of manifolds defines a multiplication <math>[M][N]=[M \times N],</math> so
:<math>\mathfrak{N}_* = \bigoplus_{n \geqslant 0}\mathfrak{N}_n</math>
is a [[graded algebra]], with the grading given by the dimension.
The cobordism class <math>[M] \in \mathfrak{N}_n</math> of a closed unoriented ''n''-dimensional manifold ''M'' is determined by the Stiefel–Whitney [[characteristic number]]s of ''M'', which depend on the stable isomorphism class of the [[tangent bundle]]. Thus if ''M'' has a stably trivial tangent bundle then <math>[M]=0 \in \mathfrak{N}_n</math>. In 1954 [[René Thom]] proved
:<math>\mathfrak{N}_* = \mathbb{F}_2 \left[x_i | i \geqslant 1, i \neq 2^j - 1 \right]</math>
the polynomial algebra with one generator <math>x_i</math> in each dimension <math>i \neq 2^j - 1</math>. Thus two unoriented closed ''n''-dimensional manifolds ''M'', ''N'' are cobordant, <math>[M] = [N] \in \mathfrak{N}_n,</math> if and only if for each collection <math>\left(i_1, \cdots, i_k\right)</math> of ''k''-tuples of integers <math>i \geqslant 1, i \neq 2^j - 1</math> such that <math>i_1 + \cdots + i_k = n</math> the Stiefel-Whitney numbers are equal
:<math>\left\langle w_{i_1}(M) \cdots w_{i_k}(M), [M] \right\rangle = \left\langle w_{i_1}(N) \cdots w_{i_k}(N), [N] \right\rangle \in \mathbb{F}_2</math>
with <math>w_i(M) \in H^i\left(M; \mathbb{F}_2\right)</math> the ''i''th [[Stiefel-Whitney class]] and <math>[M] \in H_n\left(M; \mathbb{F}_2\right)</math> the <math>\mathbb{F}_2</math>-coefficient [[fundamental class]].
For even ''i'' it is possible to choose <math>x_i = \left[\mathbb{P}^i(\R)\right]</math>, the cobordism class of the ''i''-dimensional [[real projective space]].
The low-dimensional unoriented cobordism groups are
:<math>\begin{align}
\mathfrak{N}_0 &= \Z/2, \\
\mathfrak{N}_1 &= 0, \\
\mathfrak{N}_2 &= \Z/2, \\
\mathfrak{N}_3 &= 0, \\
\mathfrak{N}_4 &= \Z/2 \oplus \Z/2, \\
\mathfrak{N}_5 &= \Z/2.
\end{align}</math>
This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).
The [[Euler characteristic]] <math>\chi(M) \in \Z</math> modulo 2 of an unoriented manifold ''M'' is an unoriented cobordism invariant. This is implied by the equation
:<math>\chi_{\partial W} = \left(1 - (-1)^{\dim W} \right)\chi_W</math>
for any compact manifold with boundary <math>W</math>.
Therefore, <math>\chi: \mathfrak{N}_i \to \Z/2</math> is a well-defined group homomorphism. For example, for any <math>i_1, \cdots, i_k \in\mathbb{N}</math>
:<math>\chi \left( \mathbb{P}^{2i_1} (\R) \times \cdots \times \mathbb{P}^{2i_k}(\R) \right) = 1.</math>
In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map <math>\chi: \mathfrak{N}_{2i} \to \Z/2</math> is onto for all <math>i \in \mathbb{N},</math> and a group isomorphism for <math>i = 1.</math>
Moreover, because of <math>\chi(M \times N) = \chi(M)\chi(N)</math>, these group homomorphism assemble into a homomorphism of graded algebras:
:<math>\begin{cases}
\mathfrak{N} \to \mathbb{F}_2[x] \\[]
[M] \mapsto \chi(M) x^{\dim(M)}
\end{cases}</math>
==Cobordism of manifolds with additional structure==
Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of ''X''-structure (or [[G-structure]]).<ref>{{Citation | last1=Switzer | first1=Robert M. | title=Algebraic topology—homotopy and homology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 | mr=1886843 | year=2002}}, chapter 12</ref> Very briefly, the [[normal bundle]] ν of an immersion of ''M'' into a sufficiently high-dimensional [[Euclidean space]] <math>\R^{n+k}</math> gives rise to a map from ''M'' to the [[Grassmannian]], which in turn is a subspace of the [[classifying space]] of the [[orthogonal group]]: ν: ''M'' → '''Gr'''(''n'', ''n'' + ''k'') → ''BO''(''k''). Given a collection of spaces and maps ''X<sub>k</sub>'' → ''X<sub>k</sub>''<sub>+1</sub> with maps ''X<sub>k</sub>'' → ''BO''(''k'') (compatible with the inclusions ''BO''(''k'') → ''BO''(''k''+1), an ''X''-structure is a lift of ν to a map <math>\tilde \nu: M \to X_k</math>. Considering only manifolds and cobordisms with ''X''-structure gives rise to a more general notion of cobordism. In particular, ''X<sub>k</sub>'' may be given by ''BG''(''k''), where ''G''(''k'') → ''O''(''k'') is some group homomorphism. This is referred to as a [[G-structure]]. Examples include ''G'' = ''O'', the orthogonal group, giving back the unoriented cobordism, but also the subgroup [[special linear group|SO(''k'')]], giving rise to [[oriented cobordism]], the [[spin group]], the [[unitary group|unitary group ''U''(''k'')]], and the trivial group, giving rise to [[framed cobordism]].
The resulting cobordism groups are then defined analogously to the unoriented case. They are denoted by <math>\Omega^G_*</math>.
===Oriented cobordism===
{{Further|List of cohomology theories#Oriented cobordism}}
Oriented cobordism is the one of manifolds with an SO-structure. Equivalently, all manifolds need to be [[orientability|oriented]] and cobordisms (''W'', ''M'', ''N'') (also referred to as ''oriented cobordisms'' for clarity) are such that the boundary (with the induced orientations) is <math>M \sqcup (-N)</math>, where −''N'' denotes ''N'' with the reversed orientation. For example, boundary of the cylinder ''M'' × ''I'' is <math>M \sqcup (-M)</math>: both ends have opposite orientations. It is also the correct definition in the sense of [[extraordinary cohomology theory]].
Unlike in the unoriented cobordism group, where every element is two-torsion, 2''M'' is not in general an oriented boundary, that is, 2[''M''] ≠ 0 when considered in <math>\Omega_*^{\text{SO}}.</math>
The oriented cobordism groups are given modulo torsion by
:<math>\Omega_*^{\text{SO}}\otimes \Q =\Q \left [y_{4i}\mid i \geqslant 1 \right ],</math>
the polynomial algebra generated by the oriented cobordism classes
:<math>y_{4i}=\left [\mathbb{P}^{2i}(\Complex) \right ] \in \Omega_{4i}^{\text{SO}}</math>
of the [[complex projective space]]s (Thom, 1952). The oriented cobordism group <math>\Omega_*^{\text{SO}}</math> is determined by the Stiefel–Whitney and Pontrjagin [[characteristic number]]s (Wall, 1960). Two oriented manifolds are oriented cobordant if and only if their Stiefel–Whitney and Pontrjagin numbers are the same.
The low-dimensional oriented cobordism groups are :
:<math>\begin{align}
\Omega_0^{\text{SO}} &= \Z, \\
\Omega_1^{\text{SO}} &= 0, \\
\Omega_2^{\text{SO}} &= 0, \\
\Omega_3^{\text{SO}} &= 0, \\
\Omega_4^{\text{SO}} &= \Z, \\
\Omega_5^{\text{SO}} &= \Z_2.
\end{align}</math>
The [[Signature of a manifold|signature]] of an oriented 4''i''-dimensional manifold ''M'' is defined as the signature of the intersection form on <math>H^{2i}(M) \in \Z</math> and is denoted by <math>\sigma(M).</math> It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the [[Hirzebruch signature theorem]].
For example, for any ''i''<sub>1</sub>, ..., ''i<sub>k</sub>'' ≥ 1
:<math>\sigma \left (\mathbb{P}^{2i_1}(\Complex) \times \cdots \times \mathbb{P}^{2i_k}(\Complex) \right) = 1.</math>
The signature map <math>\sigma:\Omega_{4i}^{\text{SO}} \to \Z</math> is onto for all ''i'' ≥ 1, and an isomorphism for ''i'' = 1.
==Cobordism as an extraordinary cohomology theory==
Every [[vector bundle]] theory (real, complex etc.) has an [[extraordinary cohomology theory]] called [[K-theory]]. Similarly, every cobordism theory Ω<sup>''G''</sup> has an [[extraordinary cohomology theory]], with homology ("bordism") groups <math>\Omega^G_n(X)</math> and cohomology ("cobordism") groups <math>\Omega^n_G(X)</math> for any space ''X''. The generalized homology groups <math>\Omega_*^G(X)</math> are [[Covariance|covariant]] in ''X'', and the generalized cohomology groups <math>\Omega^*_G(X)</math> are [[Covariance and contravariance of vectors|contravariant]] in ''X''. The cobordism groups defined above are, from this point of view, the homology groups of a point: <math>\Omega_n^G = \Omega_n^G(\text{pt})</math>. Then <math>\Omega^G_n(X)</math> is the group of ''bordism'' classes of pairs (''M'', ''f'') with ''M'' a closed ''n''-dimensional manifold ''M'' (with G-structure) and ''f'' : ''M'' → ''X'' a map. Such pairs (''M'', ''f''), (''N'', ''g'') are ''bordant'' if there exists a G-cobordism (''W''; ''M'', ''N'') with a map ''h'' : ''W'' → ''X'', which restricts to ''f'' on ''M'', and to ''g'' on ''N''.
An ''n''-dimensional manifold ''M'' has a [[Homology (mathematics)|fundamental homology class]] [''M''] ∈ ''H<sub>n</sub>''(''M'') (with coefficients in <math>\Z/2</math> in general, and in <math>\Z</math> in the oriented case), defining a natural transformation
:<math>\begin{cases}
\Omega^G_n(X) \to H_n(X) \\
(M,f) \mapsto f_*[M]
\end{cases}</math>
which is far from being an isomorphism in general.
The bordism and cobordism theories of a space satisfy the [[Eilenberg–Steenrod axioms]] apart from the dimension axiom. This does not mean that the groups <math>\Omega^n_G(X)</math> can be effectively computed once one knows the cobordism theory of a point and the homology of the space ''X'', though the [[Atiyah–Hirzebruch spectral sequence]] gives a starting point for calculations. The computation is only easy if the particular cobordism theory [[#Cobordism_as_an_extraordinary_cohomology_theory|reduces to a product of ordinary homology theories]], in which case the bordism groups are the ordinary homology groups
:<math>\Omega^G_n(X)=\sum_{p+q=n}H_p(X;\Omega^G_q(\text{pt})).</math>
This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably [[Pontrjagin–Thom construction#Framed cobordism|framed cobordism]], oriented cobordism and [[complex cobordism]]. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the [[homotopy groups of spheres]]).<ref>{{Cite book |first=D.C. |last=Ravenel |title=Complex cobordism and stable homotopy groups of spheres |publisher=Academic Press |date=April 1986 |isbn=0-12-583430-6 }}</ref>
Cobordism theories are represented by [[Thom spectrum|Thom spectra]] ''MG'': given a group ''G'', the Thom spectrum is composed from the [[Thom space]]s ''MG<sub>n</sub>'' of the [[tautological bundle|standard vector bundles]] over the [[classifying space]]s ''BG<sub>n</sub>''. Note that even for similar groups, Thom spectra can be very different: ''MSO'' and ''MO'' are very different, reflecting the difference between oriented and unoriented cobordism.
From the point of view of spectra, unoriented cobordism is a product of [[Eilenberg–MacLane spectrum|Eilenberg–MacLane spectra]] – ''MO'' = ''H''({{pi}}<sub>∗</sub>(''MO'')) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum ''MSO'' is rather more complicated than ''MO''.
==See also==
*[[h-cobordism|''h''-cobordism]]
*[[Link concordance]]
*[[List of cohomology theories]]
*[[Symplectic filling]]
*[[Cobordism hypothesis]]
*[[Cobordism ring]]
*[[Timeline of bordism]]
==Notes==
{{Reflist}}
==References==
*[[Frank Adams|John Frank Adams]], ''Stable homotopy and generalised homology'', Univ. Chicago Press (1974).
*{{springer
| title= bordism
| id=b/b017030
| last= Anosov
| first= Dmitri V.
| author-link= Dmitri Anosov
| last2= Voitsekhovskii
| first2= M. I.
| author2-link=
}}
*[[Michael Atiyah|Michael F. Atiyah]], ''Bordism and cobordism'' Proc. Camb. Phil. Soc. 57, pp. 200–208 (1961).
*{{Cite book
|first=Jean Alexandre
|last=Dieudonné
|authorlink=Jean Dieudonné
|title=A history of algebraic and differential topology, 1900–1960
|isbn=978-0-8176-3388-2
|publisher=Birkhäuser
|location=Boston
|year=1989
|url-access=registration
|url=https://archive.org/details/historyofalgebra0000dieu_g9a3
}}
*{{Cite document
|first=Antoni A.
|last=Kosinski
|title=Differential Manifolds
|publisher=Dover Publications
|date=October 19, 2007
}}
*{{Cite book
|first1=Ib
|last1=Madsen
|author1-link = Ib Madsen
|first2=R. James
|last2=Milgram
|title=The classifying spaces for surgery and cobordism of manifolds
|publisher=[[Princeton University Press]]
|location=[[Princeton, New Jersey]]
|isbn=978-0-691-08226-4
|year=1979
}}
*{{Cite journal
| issn = 0013-8584
| volume = 8
| pages = 16–23
| last = Milnor
| first = John
| authorlink=John Milnor
| title = A survey of cobordism theory
| journal = [[L'Enseignement Mathématique]]
| year = 1962
}}
*[[Sergei Novikov (mathematician)|Sergei Novikov]], ''Methods of algebraic topology from the point of view of cobordism theory'', Izv. Akad. Nauk SSSR Ser. Mat. '''31''' (1967), 855–951.
*[[Lev Pontryagin]], ''Smooth manifolds and their applications in homotopy theory'' American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959).
*[[Daniel Quillen]], ''On the formal group laws of unoriented and complex cobordism theory'' Bull. Amer. Math. Soc., 75 (1969) pp. 1293–1298.
*[[Douglas Ravenel]], ''Complex cobordism and stable homotopy groups of spheres'', Acad. Press (1986).
*{{springer|id=C/c022780|title=Cobordism|author=Yuli B. Rudyak}}
*[[Yuli Rudyak|Yuli B. Rudyak]], ''On Thom spectra, orientability, and (co)bordism'', Springer (2008).
* [[Robert Evert Stong|Robert E. Stong]], ''Notes on cobordism theory'', Princeton Univ. Press (1968).
*{{Cite book
| publisher = World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ
| isbn = 978-981-270-559-4
| volume = 39
| last = Taimanov
| first = Iskander A.
| others = [[Sergei Novikov (mathematician)|S. Novikov]] (ed.)
| title = Topological library. Part 1: cobordisms and their applications
| series = Series on Knots and Everything
| year = 2007
}}
* [[René Thom]], ''Quelques propriétés globales des variétés différentiables'', [[Commentarii Mathematici Helvetici]] 28, 17-86 (1954).
*{{Cite journal
| issn = 0003-486X
| volume = 72
| pages = 292–311
| last =Wall
| first =C. T. C.
| authorlink=C. T. C. Wall
| title = Determination of the cobordism ring
| journal = [[Annals of Mathematics]] |series=Second Series
| year = 1960
| doi = 10.2307/1970136
| issue = 2
| publisher = The Annals of Mathematics, Vol. 72, No. 2
| jstor = 1970136
}}
== External links ==
* [https://web.archive.org/web/20110719102848/http://www.map.him.uni-bonn.de/Bordism Bordism] on the Manifold Atlas.
* [https://archive.today/20120529222452/http://www.map.him.uni-bonn.de/B-Bordism B-Bordism] on the Manifold Atlas.
{{Topology}}
{{Authority control}}
[[Category:Differential topology]]
[[Category:Algebraic topology]]
[[Category:Surgery theory]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -17,35 +17,35 @@
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of <math>M</math>; the boundary of <math>M</math> is denoted by <math>\partial M</math>. Finally, a [[closed manifold]] is, by definition, a [[compact space|compact]] manifold without boundary (<math>\partial M=\emptyset</math>.)
-===Cobordisms===
-An <math>(n+1)</math>-dimensional ''cobordism'' is a [[quintuple]] <math>(W; M, N, i, j)</math> consisting of an <math>(n+1)</math>-dimensional compact differentiable manifold with boundary, <math>W</math>; closed <math>n</math>-manifolds <math>M</math>, <math>N</math>; and [[embedding]]s <math>i\colon M \hookrightarrow \partial W</math>, <math>j\colon N \hookrightarrow\partial W</math> with disjoint images such that
+===Bordisms===
+An <math>(n+1)</math>-dimensional ''bordism'' is a [[quintuple]] <math>(W; M, N, i, j)</math> consisting of an <math>(n+1)</math>-dimensional compact differentiable manifold with boundary, <math>W</math>; closed <math>n</math>-manifolds <math>M</math>, <math>N</math>; and [[embedding]]s <math>i\colon M \hookrightarrow \partial W</math>, <math>j\colon N \hookrightarrow\partial W</math> with disjoint images such that
:<math>\partial W = i(M) \sqcup j(N)~.</math>
-The terminology is usually abbreviated to <math>(W; M, N)</math>.<ref>The notation "<math>(n+1)</math>-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds.</ref> ''M'' and ''N'' are called ''cobordant'' if such a cobordism exists. All manifolds cobordant to a fixed given manifold ''M'' form the ''cobordism class'' of ''M''.
+The terminology is usually abbreviated to <math>(W; M, N)</math>.<ref>The notation "<math>(n+1)</math>-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional bordism" refers to a 5-dimensional bordism between 4-dimensional manifolds or a 6-dimensional bordism between 5-dimensional manifolds.</ref> ''M'' and ''N'' are called ''cobordant'' if such a cobordism exists. All manifolds cobordant to a fixed given manifold ''M'' form the ''bordism class'' of ''M''.
-Every closed manifold ''M'' is the boundary of the non-compact manifold ''M'' × [0, 1); for this reason we require ''W'' to be compact in the definition of cobordism. Note however that ''W'' is ''not'' required to be connected; as a consequence, if ''M'' = ∂''W''<sub>1</sub> and ''N'' = ∂''W''<sub>2</sub>, then ''M'' and ''N'' are cobordant.
+Every closed manifold ''M'' is the boundary of the non-compact manifold ''M'' × [0, 1); for this reason we require ''W'' to be compact in the definition of bordism. Note however that ''W'' is ''not'' required to be connected; as a consequence, if ''M'' = ∂''W''<sub>1</sub> and ''N'' = ∂''W''<sub>2</sub>, then ''M'' and ''N'' are cobordant.
===Examples===
-The simplest example of a cobordism is the [[unit interval]] {{nowrap|''I'' {{=}} [0, 1]}}. It is a 1-dimensional cobordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold ''M'', ({{nowrap|''M'' × ''I''}}; {{nowrap|''M'' × {0} }}, {{nowrap|''M'' × {1} }}) is a cobordism from ''M'' × {0} to ''M'' × {1}.
+The simplest example of a cobordism is the [[unit interval]] {{nowrap|''I'' {{=}} [0, 1]}}. It is a 1-dimensional bordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold ''M'', ({{nowrap|''M'' × ''I''}}; {{nowrap|''M'' × {0} }}, {{nowrap|''M'' × {1} }}) is a cobordism from ''M'' × {0} to ''M'' × {1}.
-[[File:Pair of pants cobordism (pantslike).svg|thumb|right| A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).]]
+[[File:Pair of pants bordism (pantslike).svg|thumb|right| A bordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).]]
-If ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a [[pair of pants (mathematics)|pair of pants]] ''W'' (see the figure at right). Thus the pair of pants is a cobordism between ''M'' and ''N''. A simpler cobordism between ''M'' and ''N'' is given by the disjoint union of three disks.
+If ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a [[pair of pants (mathematics)|pair of pants]] ''W'' (see the figure at right). Thus the pair of pants is a cobordism between ''M'' and ''N''. A simpler bordism between ''M'' and ''N'' is given by the disjoint union of three disks.
-The pair of pants is an example of a more general cobordism: for any two ''n''-dimensional manifolds ''M'', ''M''′, the disjoint union <math>M \sqcup M'</math> is cobordant to the [[connected sum]] <math>M\mathbin{\#}M'.</math> The previous example is a particular case, since the connected sum <math>\mathbb{S}^1\mathbin{\#}\mathbb{S}^1</math> is isomorphic to <math>\mathbb{S}^1.</math> The connected sum <math>M\mathbin{\#}M'</math> is obtained from the disjoint union <math>M \sqcup M'</math> by surgery on an embedding of <math>\mathbb{S}^0 \times \mathbb{D}^n</math> in <math>M \sqcup M'</math>, and the cobordism is the trace of the surgery.
+The pair of pants is an example of a more general bordism: for any two ''n''-dimensional manifolds ''M'', ''M''′, the disjoint union <math>M \sqcup M'</math> is cobordant to the [[connected sum]] <math>M\mathbin{\#}M'.</math> The previous example is a particular case, since the connected sum <math>\mathbb{S}^1\mathbin{\#}\mathbb{S}^1</math> is isomorphic to <math>\mathbb{S}^1.</math> The connected sum <math>M\mathbin{\#}M'</math> is obtained from the disjoint union <math>M \sqcup M'</math> by surgery on an embedding of <math>\mathbb{S}^0 \times \mathbb{D}^n</math> in <math>M \sqcup M'</math>, and the bordism is the trace of the surgery.
===Terminology===
-An ''n''-manifold ''M'' is called ''null-cobordant'' if there is a cobordism between ''M'' and the empty manifold; in other words, if ''M'' is the entire boundary of some (''n'' + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a ''n''-sphere is null-cobordant since it bounds a (''n'' + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a [[handlebody]]. On the other hand, the 2''n''-dimensional [[real projective space]] <math>\mathbb{P}^{2n}(\R)</math> is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
+An ''n''-manifold ''M'' is called ''null-cobordant'' if there is a bordism between ''M'' and the empty manifold; in other words, if ''M'' is the entire boundary of some (''n'' + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a ''n''-sphere is null-cobordant since it bounds a (''n'' + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a [[handlebody]]. On the other hand, the 2''n''-dimensional [[real projective space]] <math>\mathbb{P}^{2n}(\R)</math> is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
The general ''bordism problem'' is to calculate the cobordism classes of manifolds subject to various conditions.
-Null-cobordisms with additional structure are called [[symplectic filling|fillings]]. ''Bordism'' and ''cobordism'' are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question ''bordism of manifolds'', and the study of cobordisms as objects ''cobordisms of manifolds''.{{Citation needed|date=March 2012}}
+Null-bordisms with additional structure are called [[symplectic filling|fillings]]. ''Bordism'' and ''Cobordism'' are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question ''bordism of manifolds'', and the study of cobordisms as objects ''cobordisms of manifolds''.{{Citation needed|date=March 2012}}
-The term ''bordism'' comes from French {{lang|fr|[[wikt:bord|bord]]}}, meaning boundary. Hence bordism is the study of boundaries. ''Cobordism'' means "jointly bound", so ''M'' and ''N'' are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, cobordism groups form an extraordinary ''cohomology theory'', hence the co-.
+The term ''bordism'' comes from French {{lang|fr|[[wikt:bord|bord]]}}, meaning boundary. Hence bordism is the study of boundaries. ''Cobordism'' means "jointly bound", so ''M'' and ''N'' are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, bordism groups form an extraordinary ''cohomology theory'', hence the co-.
===Variants===
-The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are [[orientability|oriented]], or carry some other additional structure referred to as [[G-structure]]. This gives rise to [[#Oriented cobordism|"oriented cobordism"]] and "cobordism with G-structure", respectively. Under favourable technical conditions these form a [[graded ring]] called the '''cobordism ring''' <math>\Omega^G_*</math>, with grading by dimension, addition by disjoint union and multiplication by [[cartesian product]]. The cobordism groups <math>\Omega^G_*</math> are the coefficient groups of a [[#Cobordism_as_an_extraordinary_cohomology_theory|generalised homology theory]].
+The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are [[orientability|oriented]], or carry some other additional structure referred to as [[G-structure]]. This gives rise to [[#Oriented bordism|"oriented bordism"]] and "bordism with G-structure", respectively. Under favourable technical conditions these form a [[graded ring]] called the '''bordism ring''' <math>\Omega^G_*</math>, with grading by dimension, addition by disjoint union and multiplication by [[cartesian product]]. The bordism groups <math>\Omega^G_*</math> are the coefficient groups of a [[#Cobordism_as_an_extraordinary_cohomology_theory|generalised homology theory]].
-When there is additional structure, the notion of cobordism must be formulated more precisely: a ''G''-structure on ''W'' restricts to a ''G''-structure on ''M'' and ''N''. The basic examples are ''G'' = O for unoriented cobordism, ''G'' = SO for oriented cobordism, and ''G'' = U for [[complex cobordism]] using ''stably'' [[complex manifold]]s. Many more are detailed by [[Robert Evert Stong|Robert E. Stong]].<ref>{{Cite book | publisher = [[Princeton University Press]] | last = Stong | first = Robert E. | authorlink=Robert Evert Stong|title=Notes on cobordism theory |location=Princeton, NJ| year = 1968 }}</ref>
+When there is additional structure, the notion of bordism must be formulated more precisely: a ''G''-structure on ''W'' restricts to a ''G''-structure on ''M'' and ''N''. The basic examples are ''G'' = O for unoriented bordism, ''G'' = SO for oriented bordism, and ''G'' = U for [[complex bordism]] using ''stably'' [[complex manifold]]s. Many more are detailed by [[Robert Evert Stong|Robert E. Stong]].<ref>{{Cite book | publisher = [[Princeton University Press]] | last = Stong | first = Robert E. | authorlink=Robert Evert Stong|title=Notes on bordism theory |location=Princeton, NJ| year = 1968 }}</ref>
In a similar vein, a standard tool in [[surgery theory]] is surgery on [[normal invariants|normal maps]]: such a process changes a normal map to another normal map within the same [[bordism]] class.
' |
Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Cobordism.svg" class="image"><img src="/upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/220px-Cobordism.svg.png" decoding="async" width="220" height="220" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/330px-Cobordism.svg.png 1.5x, /upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/440px-Cobordism.svg.png 2x" data-file-width="200" data-file-height="200" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Cobordism.svg" class="internal" title="Enlarge"></a></div>A cobordism (<i>W</i>; <i>M</i>, <i>N</i>).</div></div></div>
<p>In <a href="/enwiki/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>cobordism</b> is a fundamental <a href="/enwiki/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on the class of <a href="/enwiki/wiki/Compact_space" title="Compact space">compact</a> <a href="/enwiki/wiki/Manifold" title="Manifold">manifolds</a> of the same dimension, set up using the concept of the <a href="/enwiki/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> (French <i><a href="https://en.wiktionary.org/wiki/bord#French" class="extiw" title="wikt:bord">bord</a></i>, giving <i>cobordism</i>) of a manifold. Two manifolds of the same dimension are <i>cobordant</i> if their <a href="/enwiki/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> is the <i>boundary</i> of a compact manifold one dimension higher.
</p><p>The boundary of an (<i>n</i> + 1)-dimensional <a href="/enwiki/wiki/Manifold" title="Manifold">manifold</a> <i>W</i> is an <i>n</i>-dimensional manifold ∂<i>W</i> that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by <a href="/enwiki/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a> for <a href="/enwiki/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a> (i.e., differentiable), but there are now also versions for
<a href="/enwiki/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise linear</a> and <a href="/enwiki/wiki/Topological_manifold" title="Topological manifold">topological manifolds</a>.
</p><p>A <i>cobordism</i> between manifolds <i>M</i> and <i>N</i> is a compact manifold <i>W</i> whose boundary is the disjoint union of <i>M</i> and <i>N</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial W=M\sqcup N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>W</mi>
<mo>=</mo>
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial W=M\sqcup N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e3a5464152cd6cd3465613eec1fa4e42449f3922" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:13.94ex; height:2.176ex;" alt="\partial W=M\sqcup N"/></span>.
</p><p>Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than <a href="/enwiki/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> or <a href="/enwiki/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to <a href="/enwiki/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> or <a href="/enwiki/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> in dimensions ≥ 4 – because the <a href="/enwiki/wiki/Word_problem_for_groups" title="Word problem for groups">word problem for groups</a> cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in <a href="/enwiki/wiki/Geometric_topology" title="Geometric topology">geometric topology</a> and <a href="/enwiki/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>. In geometric topology, cobordisms are <a href="#Connection_with_Morse_theory">intimately connected</a> with <a href="/enwiki/wiki/Morse_theory" title="Morse theory">Morse theory</a>, and <a href="/enwiki/wiki/H-cobordism" title="H-cobordism"><i>h</i>-cobordisms</a> are fundamental in the study of high-dimensional manifolds, namely <a href="/enwiki/wiki/Surgery_theory" title="Surgery theory">surgery theory</a>. In algebraic topology, cobordism theories are fundamental <a href="/enwiki/wiki/Extraordinary_cohomology_theories" class="mw-redirect" title="Extraordinary cohomology theories">extraordinary cohomology theories</a>, and <a class="mw-selflink-fragment" href="#Categorical_aspects">categories of cobordisms</a> are the domains of <a href="/enwiki/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theories</a>.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Manifolds"><span class="tocnumber">1.1</span> <span class="toctext">Manifolds</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Bordisms"><span class="tocnumber">1.2</span> <span class="toctext">Bordisms</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Examples"><span class="tocnumber">1.3</span> <span class="toctext">Examples</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Terminology"><span class="tocnumber">1.4</span> <span class="toctext">Terminology</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="#Variants"><span class="tocnumber">1.5</span> <span class="toctext">Variants</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-7"><a href="#Surgery_construction"><span class="tocnumber">2</span> <span class="toctext">Surgery construction</span></a>
<ul>
<li class="toclevel-2 tocsection-8"><a href="#Examples_2"><span class="tocnumber">2.1</span> <span class="toctext">Examples</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-9"><a href="#Morse_functions"><span class="tocnumber">3</span> <span class="toctext">Morse functions</span></a>
<ul>
<li class="toclevel-2 tocsection-10"><a href="#Geometry,_and_the_connection_with_Morse_theory_and_handlebodies"><span class="tocnumber">3.1</span> <span class="toctext">Geometry, and the connection with Morse theory and handlebodies</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-11"><a href="#History"><span class="tocnumber">4</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#Categorical_aspects"><span class="tocnumber">5</span> <span class="toctext">Categorical aspects</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Unoriented_cobordism"><span class="tocnumber">6</span> <span class="toctext">Unoriented cobordism</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#Cobordism_of_manifolds_with_additional_structure"><span class="tocnumber">7</span> <span class="toctext">Cobordism of manifolds with additional structure</span></a>
<ul>
<li class="toclevel-2 tocsection-15"><a href="#Oriented_cobordism"><span class="tocnumber">7.1</span> <span class="toctext">Oriented cobordism</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-16"><a href="#Cobordism_as_an_extraordinary_cohomology_theory"><span class="tocnumber">8</span> <span class="toctext">Cobordism as an extraordinary cohomology theory</span></a></li>
<li class="toclevel-1 tocsection-17"><a href="#See_also"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-18"><a href="#Notes"><span class="tocnumber">10</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#External_links"><span class="tocnumber">12</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Definition">Definition</span></h2>
<h3><span class="mw-headline" id="Manifolds">Manifolds</span></h3>
<p>Roughly speaking, an <i>n</i>-dimensional <a href="/enwiki/wiki/Manifold_(mathematics)" class="mw-redirect" title="Manifold (mathematics)">manifold</a> <i>M</i> is a <a href="/enwiki/wiki/Topological_space" title="Topological space">topological space</a> <a href="/enwiki/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">locally</a> (i.e., near each point) <a href="/enwiki/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to an open subset of <a href="/enwiki/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="\mathbb{R} ^{n}."/></span> A <a href="/enwiki/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">manifold with boundary</a> is similar, except that a point of <i>M</i> is allowed to have a neighborhood that is homeomorphic to an open subset of the <a href="/enwiki/wiki/Half-space_(geometry)" title="Half-space (geometry)">half-space</a>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}\mid x_{n}\geqslant 0\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}\mid x_{n}\geqslant 0\}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2aaddb379210075bbc1206fd9a74a9521f72f2ee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:29.375ex; height:2.843ex;" alt="{\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}\mid x_{n}\geqslant 0\}.}"/></span></dd></dl>
<p>Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="M"/></span>; the boundary of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="M"/></span> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8456299e2b600f44f5aa08920be090af1b35e013" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="\partial M"/></span>. Finally, a <a href="/enwiki/wiki/Closed_manifold" title="Closed manifold">closed manifold</a> is, by definition, a <a href="/enwiki/wiki/Compact_space" title="Compact space">compact</a> manifold without boundary (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M=\emptyset }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>M</mi>
<mo>=</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial M=\emptyset }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b7378cd7bbbcd5f30b043aec5472552a780add0e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:8.021ex; height:2.509ex;" alt="{\displaystyle \partial M=\emptyset }"/></span>.)
</p>
<h3><span class="mw-headline" id="Bordisms">Bordisms</span></h3>
<p>An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="(n+1)"/></span>-dimensional <i>bordism</i> is a <a href="/enwiki/wiki/Quintuple" class="mw-redirect" title="Quintuple">quintuple</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W;M,N,i,j)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>W</mi>
<mo>;</mo>
<mi>M</mi>
<mo>,</mo>
<mi>N</mi>
<mo>,</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (W;M,N,i,j)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e1c838d81aca6e63713281c2e89f1e77912089d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.647ex; height:2.843ex;" alt="{\displaystyle (W;M,N,i,j)}"/></span> consisting of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="(n+1)"/></span>-dimensional compact differentiable manifold with boundary, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="W"/></span>; closed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"/></span>-manifolds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="M"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"/></span>; and <a href="/enwiki/wiki/Embedding" title="Embedding">embeddings</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon M\hookrightarrow \partial W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">↪<!-- ↪ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>W</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\colon M\hookrightarrow \partial W}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/30912edb59d0e5985dbe9d71c96805ecfb51384a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:11.939ex; height:2.176ex;" alt="{\displaystyle i\colon M\hookrightarrow \partial W}"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon N\hookrightarrow \partial W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>:<!-- : --></mo>
<mi>N</mi>
<mo stretchy="false">↪<!-- ↪ --></mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>W</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j\colon N\hookrightarrow \partial W}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d44155758f9674edb2ef886d61678c37be33fb99" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.742ex; height:2.509ex;" alt="{\displaystyle j\colon N\hookrightarrow \partial W}"/></span> with disjoint images such that
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial W=i(M)\sqcup j(N)~.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>W</mi>
<mo>=</mo>
<mi>i</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>⊔<!-- ⊔ --></mo>
<mi>j</mi>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
<mtext> </mtext>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial W=i(M)\sqcup j(N)~.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8d2cb20651d866626abc95f5da97d09a9a03e2ba" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.547ex; height:2.843ex;" alt="\partial W=i(M)\sqcup j(N)~."/></span></dd></dl>
<p>The terminology is usually abbreviated to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W;M,N)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>W</mi>
<mo>;</mo>
<mi>M</mi>
<mo>,</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (W;M,N)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4e58e3f13ea27a29e3b8e5da29809b87dea7c8eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.818ex; height:2.843ex;" alt="{\displaystyle (W;M,N)}"/></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> <i>M</i> and <i>N</i> are called <i>cobordant</i> if such a cobordism exists. All manifolds cobordant to a fixed given manifold <i>M</i> form the <i>bordism class</i> of <i>M</i>.
</p><p>Every closed manifold <i>M</i> is the boundary of the non-compact manifold <i>M</i> × [0, 1); for this reason we require <i>W</i> to be compact in the definition of bordism. Note however that <i>W</i> is <i>not</i> required to be connected; as a consequence, if <i>M</i> = ∂<i>W</i><sub>1</sub> and <i>N</i> = ∂<i>W</i><sub>2</sub>, then <i>M</i> and <i>N</i> are cobordant.
</p>
<h3><span class="mw-headline" id="Examples">Examples</span></h3>
<p>The simplest example of a cobordism is the <a href="/enwiki/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="nowrap"><i>I</i> = [0, 1]</span>. It is a 1-dimensional bordism between the 0-dimensional manifolds {0}, {1}. More generally, for any closed manifold <i>M</i>, (<span class="nowrap"><i>M</i> × <i>I</i></span>; <span class="nowrap"><i>M</i> × {0} </span>, <span class="nowrap"><i>M</i> × {1} </span>) is a cobordism from <i>M</i> × {0} to <i>M</i> × {1}.
</p>
<div class="thumb tright"><div class="thumbinner" style="width:182px;"><a href="/enwiki//en.wikipedia.org/wiki/Special:Upload?wpDestFile=Pair_of_pants_bordism_(pantslike).svg" class="new" title="File:Pair of pants bordism (pantslike).svg">File:Pair of pants bordism (pantslike).svg</a> <div class="thumbcaption">A bordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).</div></div></div>
<p>If <i>M</i> consists of a <a href="/enwiki/wiki/Circle" title="Circle">circle</a>, and <i>N</i> of two circles, <i>M</i> and <i>N</i> together make up the boundary of a <a href="/enwiki/wiki/Pair_of_pants_(mathematics)" title="Pair of pants (mathematics)">pair of pants</a> <i>W</i> (see the figure at right). Thus the pair of pants is a cobordism between <i>M</i> and <i>N</i>. A simpler bordism between <i>M</i> and <i>N</i> is given by the disjoint union of three disks.
</p><p>The pair of pants is an example of a more general bordism: for any two <i>n</i>-dimensional manifolds <i>M</i>, <i>M</i>′, the disjoint union <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup M'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<msup>
<mi>M</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup M'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4cf83abead9f416dfc1bf4d49f96d7de9dcaa661" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.208ex; height:2.509ex;" alt="M\sqcup M'"/></span> is cobordant to the <a href="/enwiki/wiki/Connected_sum" title="Connected sum">connected sum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\mathbin {\#} M'.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mrow class="MJX-TeXAtom-BIN">
<mi mathvariant="normal">#<!-- # --></mi>
</mrow>
<msup>
<mi>M</mi>
<mo>′</mo>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\mathbin {\#} M'.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f7eb24817302b745ae7648bb0e1929bab6417a33" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.241ex; height:2.843ex;" alt="{\displaystyle M\mathbin {\#} M'.}"/></span> The previous example is a particular case, since the connected sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}\mathbin {\#} \mathbb {S} ^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-BIN">
<mi mathvariant="normal">#<!-- # --></mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{1}\mathbin {\#} \mathbb {S} ^{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7c5c6d199f9cd0af3ae13249e1888ba2cea04815" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.662ex; height:3.009ex;" alt="{\displaystyle \mathbb {S} ^{1}\mathbin {\#} \mathbb {S} ^{1}}"/></span> is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{1}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c8ca558f2933859b6f4d1d31463ca61e3699e91d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.994ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{1}.}"/></span> The connected sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\mathbin {\#} M'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mrow class="MJX-TeXAtom-BIN">
<mi mathvariant="normal">#<!-- # --></mi>
</mrow>
<msup>
<mi>M</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\mathbin {\#} M'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f9711020ae85559e111eeacde74b85568197375d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.594ex; height:2.843ex;" alt="{\displaystyle M\mathbin {\#} M'}"/></span> is obtained from the disjoint union <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup M'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<msup>
<mi>M</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup M'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4cf83abead9f416dfc1bf4d49f96d7de9dcaa661" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.208ex; height:2.509ex;" alt="M\sqcup M'"/></span> by surgery on an embedding of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2a8024ec5eff6756afcaee4db3eae49765ba5649" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.084ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{n}}"/></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup M'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<msup>
<mi>M</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup M'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4cf83abead9f416dfc1bf4d49f96d7de9dcaa661" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.208ex; height:2.509ex;" alt="M\sqcup M'"/></span>, and the bordism is the trace of the surgery.
</p>
<h3><span class="mw-headline" id="Terminology">Terminology</span></h3>
<p>An <i>n</i>-manifold <i>M</i> is called <i>null-cobordant</i> if there is a bordism between <i>M</i> and the empty manifold; in other words, if <i>M</i> is the entire boundary of some (<i>n</i> + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a <i>n</i>-sphere is null-cobordant since it bounds a (<i>n</i> + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a <a href="/enwiki/wiki/Handlebody" title="Handlebody">handlebody</a>. On the other hand, the 2<i>n</i>-dimensional <a href="/enwiki/wiki/Real_projective_space" title="Real projective space">real projective space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{2n}(\mathbb {R} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{2n}(\mathbb {R} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e6555ddaf27f606766f20f9f2446a882d4a8f466" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.948ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} ^{2n}(\mathbb {R} )}"/></span> is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
</p><p>The general <i>bordism problem</i> is to calculate the cobordism classes of manifolds subject to various conditions.
</p><p>Null-bordisms with additional structure are called <a href="/enwiki/wiki/Symplectic_filling" title="Symplectic filling">fillings</a>. <i>Bordism</i> and <i>Cobordism</i> are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question <i>bordism of manifolds</i>, and the study of cobordisms as objects <i>cobordisms of manifolds</i>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/enwiki/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2012)">citation needed</span></a></i>]</sup>
</p><p>The term <i>bordism</i> comes from French <span title="French-language text"><i lang="fr"><a href="https://en.wiktionary.org/wiki/bord" class="extiw" title="wikt:bord">bord</a></i></span>, meaning boundary. Hence bordism is the study of boundaries. <i>Cobordism</i> means "jointly bound", so <i>M</i> and <i>N</i> are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, bordism groups form an extraordinary <i>cohomology theory</i>, hence the co-.
</p>
<h3><span class="mw-headline" id="Variants">Variants</span></h3>
<p>The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are <a href="/enwiki/wiki/Orientability" title="Orientability">oriented</a>, or carry some other additional structure referred to as <a href="/enwiki/wiki/G-structure" class="mw-redirect" title="G-structure">G-structure</a>. This gives rise to <a href="#Oriented_bordism">"oriented bordism"</a> and "bordism with G-structure", respectively. Under favourable technical conditions these form a <a href="/enwiki/wiki/Graded_ring" title="Graded ring">graded ring</a> called the <b>bordism ring</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{G}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{G}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6cdd933e80e2a4d84d1465cc80c36e532e231805" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.328ex; margin-bottom: -0.344ex; width:3.202ex; height:2.843ex;" alt="\Omega _{*}^{G}"/></span>, with grading by dimension, addition by disjoint union and multiplication by <a href="/enwiki/wiki/Cartesian_product" title="Cartesian product">cartesian product</a>. The bordism groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{G}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{G}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6cdd933e80e2a4d84d1465cc80c36e532e231805" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.328ex; margin-bottom: -0.344ex; width:3.202ex; height:2.843ex;" alt="\Omega _{*}^{G}"/></span> are the coefficient groups of a <a href="#Cobordism_as_an_extraordinary_cohomology_theory">generalised homology theory</a>.
</p><p>When there is additional structure, the notion of bordism must be formulated more precisely: a <i>G</i>-structure on <i>W</i> restricts to a <i>G</i>-structure on <i>M</i> and <i>N</i>. The basic examples are <i>G</i> = O for unoriented bordism, <i>G</i> = SO for oriented bordism, and <i>G</i> = U for <a href="/enwiki/wiki/Complex_bordism" class="mw-redirect" title="Complex bordism">complex bordism</a> using <i>stably</i> <a href="/enwiki/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>. Many more are detailed by <a href="/enwiki/wiki/Robert_Evert_Stong" title="Robert Evert Stong">Robert E. Stong</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup>
</p><p>In a similar vein, a standard tool in <a href="/enwiki/wiki/Surgery_theory" title="Surgery theory">surgery theory</a> is surgery on <a href="/enwiki/wiki/Normal_invariants" class="mw-redirect" title="Normal invariants">normal maps</a>: such a process changes a normal map to another normal map within the same <a href="/enwiki/wiki/Bordism" class="mw-redirect" title="Bordism">bordism</a> class.
</p><p>Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially <a href="/enwiki/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise linear (PL)</a> and <a href="/enwiki/wiki/Topological_manifold" title="Topological manifold">topological manifolds</a>. This gives rise to bordism groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{PL}(X),\Omega _{*}^{TOP}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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<mo>,</mo>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{PL}(X),\Omega _{*}^{TOP}(X)}</annotation>
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f62e90d1ef2f6ca482b4b8a9328161ce2535a036" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.432ex; height:3.009ex;" alt="\Omega _{*}^{{PL}}(X),\Omega _{*}^{{TOP}}(X)"/></span>, which are harder to compute than the differentiable variants.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/enwiki/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2018)">citation needed</span></a></i>]</sup>
</p>
<h2><span class="mw-headline" id="Surgery_construction">Surgery construction</span></h2>
<p>Recall that in general, if <i>X</i>, <i>Y</i> are manifolds with boundary, then the boundary of the product manifold is <span class="nowrap">∂(<i>X</i> × <i>Y</i>) = (∂<i>X</i> × <i>Y</i>) ∪ (<i>X</i> × ∂<i>Y</i>)</span>.
</p><p>Now, given a manifold <i>M</i> of dimension <i>n</i> = <i>p</i> + <i>q</i> and an <a href="/enwiki/wiki/Embedding" title="Embedding">embedding</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :\mathbb {S} ^{p}\times \mathbb {D} ^{q}\subset M,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {S} ^{p}\times \mathbb {D} ^{q}\subset M,}</annotation>
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c4732f75e343c7e399b1d0d09f23aa4975cc9e5c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.504ex; height:2.843ex;" alt="{\displaystyle \varphi :\mathbb {S} ^{p}\times \mathbb {D} ^{q}\subset M,}"/></span> define the <i>n</i>-manifold
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N:=(M-\operatorname {int~im} \varphi )\cup _{\varphi |_{\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}}}\left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right)}">
<semantics>
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<mo>⁡<!-- --></mo>
<mi>φ<!-- φ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle N:=(M-\operatorname {int~im} \varphi )\cup _{\varphi |_{\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}}}\left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/79cd17b2c9d83e540cc7aaed50cb9080a03e6e98" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.505ex; width:45.296ex; height:3.843ex;" alt="{\displaystyle N:=(M-\operatorname {int~im} \varphi )\cup _{\varphi |_{\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}}}\left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right)}"/></span></dd></dl>
<p>obtained by <a href="/enwiki/wiki/Surgery_theory" title="Surgery theory">surgery</a>, via cutting out the interior of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{p}\times \mathbb {D} ^{q}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
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</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{p}\times \mathbb {D} ^{q}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4ad470408e78a8d7e7ae4d90a56016a7cf9c0d18" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.859ex; height:2.343ex;" alt="{\displaystyle \mathbb {S} ^{p}\times \mathbb {D} ^{q}}"/></span> and gluing in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
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<mi>p</mi>
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<mo>−<!-- − --></mo>
<mn>1</mn>
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</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/566bcedd5503bba2e6950901ecf5295b6a743c2f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:12.06ex; height:2.676ex;" alt="{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}}"/></span> along their boundary
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \left(\mathbb {S} ^{p}\times \mathbb {D} ^{q}\right)=\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}=\partial \left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mo>(</mo>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mo>(</mo>
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<mi mathvariant="double-struck">D</mi>
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<mo>)</mo>
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<mo>.</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial \left(\mathbb {S} ^{p}\times \mathbb {D} ^{q}\right)=\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}=\partial \left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7496415cd73d4ea77fd6beb8434b66d3e93129a3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:44.072ex; height:3.343ex;" alt="{\displaystyle \partial \left(\mathbb {S} ^{p}\times \mathbb {D} ^{q}\right)=\mathbb {S} ^{p}\times \mathbb {S} ^{q-1}=\partial \left(\mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\right).}"/></span></dd></dl>
<p>The <b>trace</b> of the surgery
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W:=(M\times I)\cup _{\mathbb {S} ^{p}\times \mathbb {D} ^{q}\times \{1\}}\left(\mathbb {D} ^{p+1}\times \mathbb {D} ^{q}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo>:=</mo>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
<mo stretchy="false">)</mo>
<msub>
<mo>∪<!-- ∪ --></mo>
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<msup>
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<annotation encoding="application/x-tex">{\displaystyle W:=(M\times I)\cup _{\mathbb {S} ^{p}\times \mathbb {D} ^{q}\times \{1\}}\left(\mathbb {D} ^{p+1}\times \mathbb {D} ^{q}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/75b55718726c71257b81397b5656c0557fc8827c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:38.471ex; height:3.509ex;" alt="{\displaystyle W:=(M\times I)\cup _{\mathbb {S} ^{p}\times \mathbb {D} ^{q}\times \{1\}}\left(\mathbb {D} ^{p+1}\times \mathbb {D} ^{q}\right)}"/></span></dd></dl>
<p>defines an <b>elementary</b> cobordism (<i>W</i>; <i>M</i>, <i>N</i>). Note that <i>M</i> is obtained from <i>N</i> by surgery on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\subset N.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\subset N.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a89f04316bc30545945f8801eec1e9a68e4d7dd4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:17.869ex; height:2.676ex;" alt="{\displaystyle \mathbb {D} ^{p+1}\times \mathbb {S} ^{q-1}\subset N.}"/></span> This is called <b>reversing the surgery</b>.
</p><p>Every cobordism is a union of elementary cobordisms, by the work of <a href="/enwiki/wiki/Marston_Morse" title="Marston Morse">Marston Morse</a>, <a href="/enwiki/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a> and <a href="/enwiki/wiki/John_Milnor" title="John Milnor">John Milnor</a>.
</p>
<h3><span class="mw-headline" id="Examples_2">Examples</span></h3>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Circle-surgery.svg" class="image"><img src="/upwiki/wikipedia/commons/thumb/f/fc/Circle-surgery.svg/220px-Circle-surgery.svg.png" decoding="async" width="220" height="100" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/f/fc/Circle-surgery.svg/330px-Circle-surgery.svg.png 1.5x, /upwiki/wikipedia/commons/thumb/f/fc/Circle-surgery.svg/440px-Circle-surgery.svg.png 2x" data-file-width="374" data-file-height="170" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Circle-surgery.svg" class="internal" title="Enlarge"></a></div>Fig. 1</div></div></div>
<p>As per the above definition, a surgery on the circle consists of cutting out a copy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>×<!-- × --></mo>
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<mi mathvariant="double-struck">D</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{1}}</annotation>
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c4a2b1c2d53f830a99303476391ded5199aa70ef" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.566ex; height:2.676ex;" alt="{\displaystyle \mathbb {D} ^{1}\times \mathbb {S} ^{0}.}"/></span> The pictures in Fig. 1 show that the result of doing this is either (i) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}}">
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</p>
<div class="thumb tleft"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Sphere-surgery1.png" class="image"><img src="/upwiki/wikipedia/commons/thumb/5/5b/Sphere-surgery1.png/220px-Sphere-surgery1.png" decoding="async" width="220" height="217" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/5/5b/Sphere-surgery1.png/330px-Sphere-surgery1.png 1.5x, /upwiki/wikipedia/commons/5/5b/Sphere-surgery1.png 2x" data-file-width="381" data-file-height="375" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Sphere-surgery1.png" class="internal" title="Enlarge"></a></div>Fig. 2a</div></div></div>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Sphere-surgery2.png" class="image"><img src="/upwiki/wikipedia/commons/thumb/4/43/Sphere-surgery2.png/220px-Sphere-surgery2.png" decoding="async" width="220" height="190" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/4/43/Sphere-surgery2.png/330px-Sphere-surgery2.png 1.5x, /upwiki/wikipedia/commons/4/43/Sphere-surgery2.png 2x" data-file-width="440" data-file-height="380" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Sphere-surgery2.png" class="internal" title="Enlarge"></a></div>Fig. 2b</div></div></div>
<p>For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}}">
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<div><ol style="list-style-type:lower-alpha"><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}\times \mathbb {D} ^{1}}">
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8264013ecdcd988a3ab4f4ece4ccc966528a2840" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{1}\times \mathbb {D} ^{1}}"/></span>: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}}">
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/65cc5747970a003100dbac17a9c8cf0a110ca0e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}}"/></span> – that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)</li><li><div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Sphere-surgery4.png" class="image"><img src="/upwiki/wikipedia/commons/thumb/4/42/Sphere-surgery4.png/220px-Sphere-surgery4.png" decoding="async" width="220" height="220" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/4/42/Sphere-surgery4.png/330px-Sphere-surgery4.png 1.5x, /upwiki/wikipedia/commons/thumb/4/42/Sphere-surgery4.png/440px-Sphere-surgery4.png 2x" data-file-width="720" data-file-height="720" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Sphere-surgery4.png" class="internal" title="Enlarge"></a></div>Fig. 2c. This shape cannot be embedded in 3-space.</div></div></div>
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}}">
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/65cc5747970a003100dbac17a9c8cf0a110ca0e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}}"/></span>: Having cut out two disks <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2},}">
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/17f5cfc20fd87a6858f7c7ff176ca5d2341800e5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.566ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{1}\times \mathbb {D} ^{1}.}"/></span> There are two possible outcomes, depending on whether our gluing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the <a href="/enwiki/wiki/Torus" title="Torus">torus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}\times \mathbb {S} ^{1}}">
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<h2><span class="mw-headline" id="Morse_functions">Morse functions</span></h2>
<p>Suppose that <i>f</i> is a <a href="/enwiki/wiki/Morse_function" class="mw-redirect" title="Morse function">Morse function</a> on an (<i>n</i> + 1)-dimensional manifold, and suppose that <i>c</i> is a critical value with exactly one critical point in its preimage. If the index of this critical point is <i>p</i> + 1, then the level-set <i>N</i> := <i>f</i><sup>−1</sup>(<i>c</i> + ε) is obtained from <i>M</i> := <i>f</i><sup>−1</sup>(<i>c</i> − ε) by a <i>p</i>-surgery. The inverse image <i>W</i> := <i>f</i><sup>−1</sup>([<i>c</i> − ε, <i>c</i> + ε]) defines a cobordism (<i>W</i>; <i>M</i>, <i>N</i>) that can be identified with the trace of this surgery.
</p>
<h3><span id="Geometry.2C_and_the_connection_with_Morse_theory_and_handlebodies"></span><span class="mw-headline" id="Geometry,_and_the_connection_with_Morse_theory_and_handlebodies">Geometry, and the connection with Morse theory and handlebodies</span></h3>
<p>Given a cobordism (<i>W</i>; <i>M</i>, <i>N</i>) there exists a smooth function <i>f</i> : <i>W</i> → [0, 1] such that <i>f</i><sup>−1</sup>(0) = <i>M</i>, <i>f</i><sup>−1</sup>(1) = <i>N</i>. By general position, one can assume <i>f</i> is Morse and such that all critical points occur in the interior of <i>W</i>. In this setting <i>f</i> is called a Morse function on a cobordism. The cobordism (<i>W</i>; <i>M</i>, <i>N</i>) is a union of the traces of a sequence of surgeries on <i>M</i>, one for each critical point of <i>f</i>. The manifold <i>W</i> is obtained from <i>M</i> × [0, 1] by attaching one <a href="/enwiki/wiki/Handle_decomposition" title="Handle decomposition">handle</a> for each critical point of <i>f</i>.
</p>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Cobordism.svg" class="image"><img src="/upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/220px-Cobordism.svg.png" decoding="async" width="220" height="220" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/330px-Cobordism.svg.png 1.5x, /upwiki/wikipedia/commons/thumb/4/49/Cobordism.svg/440px-Cobordism.svg.png 2x" data-file-width="200" data-file-height="200" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Cobordism.svg" class="internal" title="Enlarge"></a></div>The 3-dimensional cobordism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mathbb {S} ^{1}\times \mathbb {D} ^{2}-\mathbb {D} ^{3}}">
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<annotation encoding="application/x-tex">{\displaystyle M=\mathbb {S} ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ddc478a97076ad96247669b57639c8fbf8481b9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.887ex; height:2.676ex;" alt="{\displaystyle M=\mathbb {S} ^{2}}"/></span> and the 2-<a href="/enwiki/wiki/Torus" title="Torus">torus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\mathbb {S} ^{1}\times \mathbb {S} ^{1},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N=\mathbb {S} ^{1}\times \mathbb {S} ^{1},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c91588e907addab4a08800e91ab74a9fa56c7451" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.343ex; height:3.009ex;" alt="{\displaystyle N=\mathbb {S} ^{1}\times \mathbb {S} ^{1},}"/></span> with <i>N</i> obtained from <i>M</i> by surgery on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}\subset M,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">S</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⊂<!-- ⊂ --></mo>
<mi>M</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}\subset M,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/171ab04d2226f32122083b7ce8d42c871661b9ad" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:14.107ex; height:3.009ex;" alt="{\displaystyle \mathbb {S} ^{0}\times \mathbb {D} ^{2}\subset M,}"/></span>and <i>W</i> obtained from <i>M</i> × <i>I</i> by attaching a 1-handle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{1}\times \mathbb {D} ^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">D</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{1}\times \mathbb {D} ^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/786a5e8f193e177fd62aebf7e2a84fa9be7953a1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.952ex; height:2.676ex;" alt="{\displaystyle \mathbb {D} ^{1}\times \mathbb {D} ^{2}.}"/></span></div></div></div>
<p>The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of <i>f</i>′ give rise to a <a href="/enwiki/wiki/Handle_decomposition" title="Handle decomposition">handle presentation</a> of the triple (<i>W</i>; <i>M</i>, <i>N</i>). Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.
</p>
<h2><span class="mw-headline" id="History">History</span></h2>
<p>Cobordism had its roots in the (failed) attempt by <a href="/enwiki/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> in 1895 to define <a href="/enwiki/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> purely in terms of manifolds (<a href="#CITEREFDieudonné1989">Dieudonné 1989</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/historyofalgebra0000dieu_g9a3/page/290">p. 289</a>). Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See <a href="#Cobordism_as_an_extraordinary_cohomology_theory">Cobordism as an extraordinary cohomology theory</a> for the relationship between bordism and homology.
</p><p>Bordism was explicitly introduced by <a href="/enwiki/wiki/Lev_Pontryagin" title="Lev Pontryagin">Lev Pontryagin</a> in geometric work on manifolds. It came to prominence when <a href="/enwiki/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a> showed that cobordism groups could be computed by means of <a href="/enwiki/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a>, via the <a href="/enwiki/wiki/Thom_complex" class="mw-redirect" title="Thom complex">Thom complex</a> construction. Cobordism theory became part of the apparatus of <a href="/enwiki/wiki/Extraordinary_cohomology_theory" class="mw-redirect" title="Extraordinary cohomology theory">extraordinary cohomology theory</a>, alongside <a href="/enwiki/wiki/K-theory" title="K-theory">K-theory</a>. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the <a href="/enwiki/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" title="Hirzebruch–Riemann–Roch theorem">Hirzebruch–Riemann–Roch theorem</a>, and in the first proofs of the <a href="/enwiki/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>.
</p><p>In the 1980s the <a href="/enwiki/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> with compact manifolds as <a href="/enwiki/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> and cobordisms between these as <a href="/enwiki/wiki/Morphism" title="Morphism">morphisms</a> played a basic role in the Atiyah–Segal axioms for <a href="/enwiki/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a>, which is an important part of <a href="/enwiki/wiki/Quantum_topology" title="Quantum topology">quantum topology</a>.
</p>
<h2><span class="mw-headline" id="Categorical_aspects">Categorical aspects</span></h2>
<p>Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a <a href="/enwiki/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of (<i>W</i>; <i>M</i>, <i>N</i>) and (<i>W</i> ′; <i>N</i>, <i>P</i>) is defined by gluing the right end of the first to the left end of the second, yielding (<i>W</i> ′ ∪<sub><i>N</i></sub> <i>W</i>; <i>M</i>, <i>P</i>). A cobordism is a kind of <a href="/enwiki/wiki/Cospan" class="mw-redirect" title="Cospan">cospan</a>:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> <i>M</i> → <i>W</i> ← <i>N</i>. The category is a <a href="/enwiki/wiki/Dagger_compact_category" title="Dagger compact category">dagger compact category</a>.
</p><p>A <a href="/enwiki/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a> is a <a href="/enwiki/wiki/Monoidal_functor" title="Monoidal functor">monoidal functor</a> from a category of cobordisms to a category of <a href="/enwiki/wiki/Vector_space" title="Vector space">vector spaces</a>. That is, it is a <a href="/enwiki/wiki/Functor" title="Functor">functor</a> whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
</p><p>In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a nullary (0-ary) operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
</p>
<h2><span class="mw-headline" id="Unoriented_cobordism">Unoriented cobordism</span></h2>
<style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/enwiki/wiki/List_of_cohomology_theories#Unoriented_cobordism" title="List of cohomology theories">List of cohomology theories § Unoriented cobordism</a></div>
<p>The set of cobordism classes of closed unoriented <i>n</i>-dimensional manifolds is usually denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d859d3e73371381b5d9a6408966d546e4dbd686e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.152ex; height:2.509ex;" alt="{\mathfrak {N}}_{n}"/></span> (rather than the more systematic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{n}^{\text{O}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>O</mtext>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{n}^{\text{O}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d14469513cb895f4864ee01c2c7d3361465cd2fa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.843ex;" alt="\Omega _{n}^{{{\text{O}}}}"/></span>); it is an <a href="/enwiki/wiki/Abelian_group" title="Abelian group">abelian group</a> with the disjoint union as operation. More specifically, if [<i>M</i>] and [<i>N</i>] denote the cobordism classes of the manifolds <i>M</i> and <i>N</i> respectively, we define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]+[N]=[M\sqcup N]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]+[N]=[M\sqcup N]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d5df1c8215ebc8e5895096c4d8dad3c253c41d54" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.414ex; height:2.843ex;" alt="[M]+[N]=[M\sqcup N]"/></span>; this is a well-defined operation which turns <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d859d3e73371381b5d9a6408966d546e4dbd686e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.152ex; height:2.509ex;" alt="{\mathfrak {N}}_{n}"/></span> into an abelian group. The identity element of this group is the class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\emptyset ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\emptyset ]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e508e41134570bb612d4b7f85331dbd1131f43f1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="[\emptyset ]"/></span> consisting of all closed <i>n</i>-manifolds which are boundaries. Further we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]+[M]=[\emptyset ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]+[M]=[\emptyset ]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/453b464d2065d6ed8ad85c66e6ab685a4e600af4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.867ex; height:2.843ex;" alt="[M]+[M]=[\emptyset ]"/></span> for every <i>M</i> since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup M=\partial (M\times [0,1])}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<mi>M</mi>
<mo>=</mo>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>×<!-- × --></mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup M=\partial (M\times [0,1])}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/54882c4e0364c0004abc68327d51df36792fd54d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:23.628ex; height:2.843ex;" alt="{\displaystyle M\sqcup M=\partial (M\times [0,1])}"/></span>. Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d859d3e73371381b5d9a6408966d546e4dbd686e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.152ex; height:2.509ex;" alt="{\mathfrak {N}}_{n}"/></span> is a vector space over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="\mathbb {F} _{2}"/></span>, the <a href="/enwiki/wiki/GF(2)" title="GF(2)">field with two elements</a>. The cartesian product of manifolds defines a multiplication <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M][N]=[M\times N],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>×<!-- × --></mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M][N]=[M\times N],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/968cb102dae2cd6ab5f0d04a19a8bb34280f0d1a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.478ex; height:2.843ex;" alt="{\displaystyle [M][N]=[M\times N],}"/></span> so
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{*}=\bigoplus _{n\geqslant 0}{\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>⨁<!-- ⨁ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{*}=\bigoplus _{n\geqslant 0}{\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e1e22ca1d0bca86cb4322ecd81a0799e77d20c62" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:13.136ex; height:5.676ex;" alt="{\displaystyle {\mathfrak {N}}_{*}=\bigoplus _{n\geqslant 0}{\mathfrak {N}}_{n}}"/></span></dd></dl>
<p>is a <a href="/enwiki/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">graded algebra</a>, with the grading given by the dimension.
</p><p>The cobordism class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]\in {\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]\in {\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/688ae41923d2c065bd498885eb078610a62ef917" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.729ex; height:2.843ex;" alt="[M]\in {\mathfrak {N}}_{n}"/></span> of a closed unoriented <i>n</i>-dimensional manifold <i>M</i> is determined by the Stiefel–Whitney <a href="/enwiki/wiki/Characteristic_number" class="mw-redirect" title="Characteristic number">characteristic numbers</a> of <i>M</i>, which depend on the stable isomorphism class of the <a href="/enwiki/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a>. Thus if <i>M</i> has a stably trivial tangent bundle then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]=0\in {\mathfrak {N}}_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0</mn>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]=0\in {\mathfrak {N}}_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c2ca3e202d48555ee3e6d2eb3d2adeebf22da241" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:2.843ex;" alt="[M]=0\in {\mathfrak {N}}_{n}"/></span>. In 1954 <a href="/enwiki/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a> proved
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{*}=\mathbb {F} _{2}\left[x_{i}|i\geqslant 1,i\neq 2^{j}-1\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mrow>
<mo>[</mo>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>i</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>i</mi>
<mo>≠<!-- ≠ --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{*}=\mathbb {F} _{2}\left[x_{i}|i\geqslant 1,i\neq 2^{j}-1\right]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6ec625b293ad41e34b475297c8c8685ecdb75176" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:29.737ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {N}}_{*}=\mathbb {F} _{2}\left[x_{i}|i\geqslant 1,i\neq 2^{j}-1\right]}"/></span></dd></dl>
<p>the polynomial algebra with one generator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="x_{i}"/></span> in each dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq 2^{j}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>≠<!-- ≠ --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\neq 2^{j}-1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2d45898af1f5efcfea91e3da6d2d339a1e8f0e67" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.976ex; height:3.176ex;" alt="{\displaystyle i\neq 2^{j}-1}"/></span>. Thus two unoriented closed <i>n</i>-dimensional manifolds <i>M</i>, <i>N</i> are cobordant, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]=[N]\in {\mathfrak {N}}_{n},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]=[N]\in {\mathfrak {N}}_{n},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/867180f60939caa7954d902db704b311611011d9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.831ex; height:2.843ex;" alt="{\displaystyle [M]=[N]\in {\mathfrak {N}}_{n},}"/></span> if and only if for each collection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(i_{1},\cdots ,i_{k}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>,</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left(i_{1},\cdots ,i_{k}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4a88a620d3f38bb3d9702c504786f238ef7f292f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.735ex; height:2.843ex;" alt="{\displaystyle \left(i_{1},\cdots ,i_{k}\right)}"/></span> of <i>k</i>-tuples of integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geqslant 1,i\neq 2^{j}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>i</mi>
<mo>≠<!-- ≠ --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\geqslant 1,i\neq 2^{j}-1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4613cb13af6269e3d8708feaf4a2c49492692e87" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.073ex; height:3.176ex;" alt="{\displaystyle i\geqslant 1,i\neq 2^{j}-1}"/></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1}+\cdots +i_{k}=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{1}+\cdots +i_{k}=n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ee3a8655c0e25110c533461b78fb095ce7b61c90" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:16.645ex; height:2.509ex;" alt="{\displaystyle i_{1}+\cdots +i_{k}=n}"/></span> the Stiefel-Whitney numbers are equal
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle w_{i_{1}}(M)\cdots w_{i_{k}}(M),[M]\right\rangle =\left\langle w_{i_{1}}(N)\cdots w_{i_{k}}(N),[N]\right\rangle \in \mathbb {F} _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>⟨</mo>
<mrow>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>⟩</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>⟨</mo>
<mrow>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>⟩</mo>
</mrow>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\langle w_{i_{1}}(M)\cdots w_{i_{k}}(M),[M]\right\rangle =\left\langle w_{i_{1}}(N)\cdots w_{i_{k}}(N),[N]\right\rangle \in \mathbb {F} _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/01d6c65bbd1151567cbf533aaba8d75cb878564b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:58.446ex; height:3.176ex;" alt="{\displaystyle \left\langle w_{i_{1}}(M)\cdots w_{i_{k}}(M),[M]\right\rangle =\left\langle w_{i_{1}}(N)\cdots w_{i_{k}}(N),[N]\right\rangle \in \mathbb {F} _{2}}"/></span></dd></dl>
<p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}(M)\in H^{i}\left(M;\mathbb {F} _{2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>M</mi>
<mo>;</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{i}(M)\in H^{i}\left(M;\mathbb {F} _{2}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/513cf59c5a0bae1edda6a1781f5a8c92194492c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.606ex; height:3.176ex;" alt="{\displaystyle w_{i}(M)\in H^{i}\left(M;\mathbb {F} _{2}\right)}"/></span> the <i>i</i>th <a href="/enwiki/wiki/Stiefel-Whitney_class" class="mw-redirect" title="Stiefel-Whitney class">Stiefel-Whitney class</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]\in H_{n}\left(M;\mathbb {F} _{2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo>∈<!-- ∈ --></mo>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mi>M</mi>
<mo>;</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [M]\in H_{n}\left(M;\mathbb {F} _{2}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/cf7cdb819ff58f76a803179a5bfb7da600aa4ead" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.873ex; height:2.843ex;" alt="{\displaystyle [M]\in H_{n}\left(M;\mathbb {F} _{2}\right)}"/></span> the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="\mathbb {F} _{2}"/></span>-coefficient <a href="/enwiki/wiki/Fundamental_class" title="Fundamental class">fundamental class</a>.
</p><p>For even <i>i</i> it is possible to choose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\left[\mathbb {P} ^{i}(\mathbb {R} )\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}=\left[\mathbb {P} ^{i}(\mathbb {R} )\right]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d59b67aebb9b919315aefb9efb952e69b45c1cb0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:12.874ex; height:3.343ex;" alt="{\displaystyle x_{i}=\left[\mathbb {P} ^{i}(\mathbb {R} )\right]}"/></span>, the cobordism class of the <i>i</i>-dimensional <a href="/enwiki/wiki/Real_projective_space" title="Real projective space">real projective space</a>.
</p><p>The low-dimensional unoriented cobordism groups are
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathfrak {N}}_{0}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{1}&=0,\\{\mathfrak {N}}_{2}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{3}&=0,\\{\mathfrak {N}}_{4}&=\mathbb {Z} /2\oplus \mathbb {Z} /2,\\{\mathfrak {N}}_{5}&=\mathbb {Z} /2.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>⊕<!-- ⊕ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2.</mn>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathfrak {N}}_{0}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{1}&=0,\\{\mathfrak {N}}_{2}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{3}&=0,\\{\mathfrak {N}}_{4}&=\mathbb {Z} /2\oplus \mathbb {Z} /2,\\{\mathfrak {N}}_{5}&=\mathbb {Z} /2.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/eb19b1dd7883e739704f986049632a0d4dc99700" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.671ex; width:18.075ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\mathfrak {N}}_{0}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{1}&=0,\\{\mathfrak {N}}_{2}&=\mathbb {Z} /2,\\{\mathfrak {N}}_{3}&=0,\\{\mathfrak {N}}_{4}&=\mathbb {Z} /2\oplus \mathbb {Z} /2,\\{\mathfrak {N}}_{5}&=\mathbb {Z} /2.\end{aligned}}}"/></span></dd></dl>
<p>This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).
</p><p>The <a href="/enwiki/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (M)\in \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi (M)\in \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/463d8f4c38a71798c45639dca70d2f3cbb06674b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.098ex; height:2.843ex;" alt="{\displaystyle \chi (M)\in \mathbb {Z} }"/></span> modulo 2 of an unoriented manifold <i>M</i> is an unoriented cobordism invariant. This is implied by the equation
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\partial W}=\left(1-(-1)^{\dim W}\right)\chi _{W}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>W</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>dim</mi>
<mo>⁡<!-- --></mo>
<mi>W</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>W</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi _{\partial W}=\left(1-(-1)^{\dim W}\right)\chi _{W}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/10e13fcbf22487b791ddfeafa0ad80f92f4952a5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:27.23ex; height:3.343ex;" alt="{\displaystyle \chi _{\partial W}=\left(1-(-1)^{\dim W}\right)\chi _{W}}"/></span></dd></dl>
<p>for any compact manifold with boundary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="W"/></span>.
</p><p>Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :{\mathfrak {N}}_{i}\to \mathbb {Z} /2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo>:</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi :{\mathfrak {N}}_{i}\to \mathbb {Z} /2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/44d8998d9fa9284f71cbece20bb49bf0fe2f1986" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.615ex; height:2.843ex;" alt="{\displaystyle \chi :{\mathfrak {N}}_{i}\to \mathbb {Z} /2}"/></span> is a well-defined group homomorphism. For example, for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1},\cdots ,i_{k}\in \mathbb {N} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>,</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{1},\cdots ,i_{k}\in \mathbb {N} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9a5c81eb8536d98736c7a25f306d3638894dd651" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.445ex; height:2.509ex;" alt="{\displaystyle i_{1},\cdots ,i_{k}\in \mathbb {N} }"/></span>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi \left(\mathbb {P} ^{2i_{1}}(\mathbb {R} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {R} )\right)=1.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi \left(\mathbb {P} ^{2i_{1}}(\mathbb {R} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {R} )\right)=1.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/657dff98a4aa9766f1017a09f47ebe3ed0b51d65" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:32.034ex; height:3.343ex;" alt="{\displaystyle \chi \left(\mathbb {P} ^{2i_{1}}(\mathbb {R} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {R} )\right)=1.}"/></span></dd></dl>
<p>In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :{\mathfrak {N}}_{2i}\to \mathbb {Z} /2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo>:</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi :{\mathfrak {N}}_{2i}\to \mathbb {Z} /2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/27027211e532ad3f9df67a2bb9493fab851a589d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.437ex; height:2.843ex;" alt="{\displaystyle \chi :{\mathfrak {N}}_{2i}\to \mathbb {Z} /2}"/></span> is onto for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\in \mathbb {N} ,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2ebc93b067b01dc98b77de4a3411ccf9f93bf181" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.968ex; height:2.509ex;" alt="{\displaystyle i\in \mathbb {N} ,}"/></span> and a group isomorphism for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mn>1.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i=1.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e8236fc04c7aa70652114f95dfada9a917b876ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.71ex; height:2.176ex;" alt="{\displaystyle i=1.}"/></span>
</p><p>Moreover, because of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (M\times N)=\chi (M)\chi (N)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>×<!-- × --></mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi (M\times N)=\chi (M)\chi (N)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9e7e1910db91c0a80b00af4d55233d2cb1b26443" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:24.744ex; height:2.843ex;" alt="{\displaystyle \chi (M\times N)=\chi (M)\chi (N)}"/></span>, these group homomorphism assemble into a homomorphism of graded algebras:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\mathfrak {N}}\to \mathbb {F} _{2}[x]\\[][M]\mapsto \chi (M)x^{\dim(M)}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">N</mi>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>dim</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\mathfrak {N}}\to \mathbb {F} _{2}[x]\\[][M]\mapsto \chi (M)x^{\dim(M)}\end{cases}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6a2523d1744edd17a3a19fb228fecc17caab77b7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:22.86ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}{\mathfrak {N}}\to \mathbb {F} _{2}[x]\\[][M]\mapsto \chi (M)x^{\dim(M)}\end{cases}}}"/></span></dd></dl>
<h2><span class="mw-headline" id="Cobordism_of_manifolds_with_additional_structure">Cobordism of manifolds with additional structure</span></h2>
<p>Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of <i>X</i>-structure (or <a href="/enwiki/wiki/G-structure" class="mw-redirect" title="G-structure">G-structure</a>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> Very briefly, the <a href="/enwiki/wiki/Normal_bundle" title="Normal bundle">normal bundle</a> ν of an immersion of <i>M</i> into a sufficiently high-dimensional <a href="/enwiki/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n+k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n+k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e3d4cf8e6d46e93074ae9c4f5a208199596fe252" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.032ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n+k}}"/></span> gives rise to a map from <i>M</i> to the <a href="/enwiki/wiki/Grassmannian" title="Grassmannian">Grassmannian</a>, which in turn is a subspace of the <a href="/enwiki/wiki/Classifying_space" title="Classifying space">classifying space</a> of the <a href="/enwiki/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>: ν: <i>M</i> → <b>Gr</b>(<i>n</i>, <i>n</i> + <i>k</i>) → <i>BO</i>(<i>k</i>). Given a collection of spaces and maps <i>X<sub>k</sub></i> → <i>X<sub>k</sub></i><sub>+1</sub> with maps <i>X<sub>k</sub></i> → <i>BO</i>(<i>k</i>) (compatible with the inclusions <i>BO</i>(<i>k</i>) → <i>BO</i>(<i>k</i>+1), an <i>X</i>-structure is a lift of ν to a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}:M\to X_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>:</mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}:M\to X_{k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e3a91e5fccc6ec5adda17b1aa5be9cfd3ec3c749" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:12.323ex; height:2.509ex;" alt="{\tilde \nu }:M\to X_{k}"/></span>. Considering only manifolds and cobordisms with <i>X</i>-structure gives rise to a more general notion of cobordism. In particular, <i>X<sub>k</sub></i> may be given by <i>BG</i>(<i>k</i>), where <i>G</i>(<i>k</i>) → <i>O</i>(<i>k</i>) is some group homomorphism. This is referred to as a <a href="/enwiki/wiki/G-structure" class="mw-redirect" title="G-structure">G-structure</a>. Examples include <i>G</i> = <i>O</i>, the orthogonal group, giving back the unoriented cobordism, but also the subgroup <a href="/enwiki/wiki/Special_linear_group" title="Special linear group">SO(<i>k</i>)</a>, giving rise to <a href="/enwiki/wiki/Oriented_cobordism" class="mw-redirect" title="Oriented cobordism">oriented cobordism</a>, the <a href="/enwiki/wiki/Spin_group" title="Spin group">spin group</a>, the <a href="/enwiki/wiki/Unitary_group" title="Unitary group">unitary group <i>U</i>(<i>k</i>)</a>, and the trivial group, giving rise to <a href="/enwiki/w/index.php?title=Framed_cobordism&action=edit&redlink=1" class="new" title="Framed cobordism (page does not exist)">framed cobordism</a>.
</p><p>The resulting cobordism groups are then defined analogously to the unoriented case. They are denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{G}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{G}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6cdd933e80e2a4d84d1465cc80c36e532e231805" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.328ex; margin-bottom: -0.344ex; width:3.202ex; height:2.843ex;" alt="\Omega _{*}^{G}"/></span>.
</p>
<h3><span class="mw-headline" id="Oriented_cobordism">Oriented cobordism</span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"/><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/enwiki/wiki/List_of_cohomology_theories#Oriented_cobordism" title="List of cohomology theories">List of cohomology theories § Oriented cobordism</a></div>
<p>Oriented cobordism is the one of manifolds with an SO-structure. Equivalently, all manifolds need to be <a href="/enwiki/wiki/Orientability" title="Orientability">oriented</a> and cobordisms (<i>W</i>, <i>M</i>, <i>N</i>) (also referred to as <i>oriented cobordisms</i> for clarity) are such that the boundary (with the induced orientations) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup (-N)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup (-N)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/751c30ea69b87ae2ca0c821941c659b46f6e598f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.706ex; height:2.843ex;" alt="M\sqcup (-N)"/></span>, where −<i>N</i> denotes <i>N</i> with the reversed orientation. For example, boundary of the cylinder <i>M</i> × <i>I</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\sqcup (-M)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>⊔<!-- ⊔ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M\sqcup (-M)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6a3d2852134958412314849c5ebe909202403c24" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.084ex; height:2.843ex;" alt="M\sqcup (-M)"/></span>: both ends have opposite orientations. It is also the correct definition in the sense of <a href="/enwiki/wiki/Extraordinary_cohomology_theory" class="mw-redirect" title="Extraordinary cohomology theory">extraordinary cohomology theory</a>.
</p><p>Unlike in the unoriented cobordism group, where every element is two-torsion, 2<i>M</i> is not in general an oriented boundary, that is, 2[<i>M</i>] ≠ 0 when considered in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{\text{SO}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{\text{SO}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d6bc111bd84ad161117e155473033ed47205d595" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.328ex; margin-bottom: -0.344ex; width:4.75ex; height:2.843ex;" alt="\Omega _{*}^{{{\text{SO}}}}."/></span>
</p><p>The oriented cobordism groups are given modulo torsion by
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} =\mathbb {Q} \left[y_{4i}\mid i\geqslant 1\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>∣<!-- ∣ --></mo>
<mi>i</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>1</mn>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} =\mathbb {Q} \left[y_{4i}\mid i\geqslant 1\right],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/252346a8b3d557f0d9e9982d1312d11eee58bc28" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:26.134ex; height:3.009ex;" alt="{\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} =\mathbb {Q} \left[y_{4i}\mid i\geqslant 1\right],}"/></span></dd></dl>
<p>the polynomial algebra generated by the oriented cobordism classes
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{4i}=\left[\mathbb {P} ^{2i}(\mathbb {C} )\right]\in \Omega _{4i}^{\text{SO}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>∈<!-- ∈ --></mo>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{4i}=\left[\mathbb {P} ^{2i}(\mathbb {C} )\right]\in \Omega _{4i}^{\text{SO}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/67c79c9b9ad5de85b731f3bd4594e6d6aef751fc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:21.271ex; height:3.343ex;" alt="{\displaystyle y_{4i}=\left[\mathbb {P} ^{2i}(\mathbb {C} )\right]\in \Omega _{4i}^{\text{SO}}}"/></span></dd></dl>
<p>of the <a href="/enwiki/wiki/Complex_projective_space" title="Complex projective space">complex projective spaces</a> (Thom, 1952). The oriented cobordism group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{\text{SO}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{\text{SO}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/958c934cf0019ffc980a74fcf209a7c72f67b09d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.328ex; margin-bottom: -0.344ex; width:4.103ex; height:2.843ex;" alt="\Omega _{*}^{{{\text{SO}}}}"/></span> is determined by the Stiefel–Whitney and Pontrjagin <a href="/enwiki/wiki/Characteristic_number" class="mw-redirect" title="Characteristic number">characteristic numbers</a> (Wall, 1960). Two oriented manifolds are oriented cobordant if and only if their Stiefel–Whitney and Pontrjagin numbers are the same.
</p><p>The low-dimensional oriented cobordism groups are :
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Omega _{0}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{1}^{\text{SO}}&=0,\\\Omega _{2}^{\text{SO}}&=0,\\\Omega _{3}^{\text{SO}}&=0,\\\Omega _{4}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{5}^{\text{SO}}&=\mathbb {Z} _{2}.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Omega _{0}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{1}^{\text{SO}}&=0,\\\Omega _{2}^{\text{SO}}&=0,\\\Omega _{3}^{\text{SO}}&=0,\\\Omega _{4}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{5}^{\text{SO}}&=\mathbb {Z} _{2}.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f4cd987078eca23c47a8a2e0c082619f1f467ae1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -9.366ex; margin-bottom: -0.306ex; width:11.204ex; height:20.509ex;" alt="{\displaystyle {\begin{aligned}\Omega _{0}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{1}^{\text{SO}}&=0,\\\Omega _{2}^{\text{SO}}&=0,\\\Omega _{3}^{\text{SO}}&=0,\\\Omega _{4}^{\text{SO}}&=\mathbb {Z} ,\\\Omega _{5}^{\text{SO}}&=\mathbb {Z} _{2}.\end{aligned}}}"/></span></dd></dl>
<p>The <a href="/enwiki/wiki/Signature_of_a_manifold" class="mw-redirect" title="Signature of a manifold">signature</a> of an oriented 4<i>i</i>-dimensional manifold <i>M</i> is defined as the signature of the intersection form on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2i}(M)\in \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H^{2i}(M)\in \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f335c8726ae27198a57f9204e66824ecfeb4c7d9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.368ex; height:3.176ex;" alt="{\displaystyle H^{2i}(M)\in \mathbb {Z} }"/></span> and is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (M).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>σ<!-- σ --></mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma (M).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a1e5b0211555a449729e123ab5d194176424f762" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.228ex; height:2.843ex;" alt="{\displaystyle \sigma (M).}"/></span> It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the <a href="/enwiki/wiki/Hirzebruch_signature_theorem" title="Hirzebruch signature theorem">Hirzebruch signature theorem</a>.
</p><p>For example, for any <i>i</i><sub>1</sub>, ..., <i>i<sub>k</sub></i> ≥ 1
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \left(\mathbb {P} ^{2i_{1}}(\mathbb {C} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {C} )\right)=1.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>σ<!-- σ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma \left(\mathbb {P} ^{2i_{1}}(\mathbb {C} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {C} )\right)=1.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/50f6e428020fda191d7bbbda96b89d900d402bfa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:31.908ex; height:3.343ex;" alt="{\displaystyle \sigma \left(\mathbb {P} ^{2i_{1}}(\mathbb {C} )\times \cdots \times \mathbb {P} ^{2i_{k}}(\mathbb {C} )\right)=1.}"/></span></dd></dl>
<p>The signature map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma :\Omega _{4i}^{\text{SO}}\to \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>σ<!-- σ --></mi>
<mo>:</mo>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>SO</mtext>
</mrow>
</msubsup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma :\Omega _{4i}^{\text{SO}}\to \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9ada4de495489c285d95c10f40b3a49142546e76" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:12.534ex; height:3.176ex;" alt="{\displaystyle \sigma :\Omega _{4i}^{\text{SO}}\to \mathbb {Z} }"/></span> is onto for all <i>i</i> ≥ 1, and an isomorphism for <i>i</i> = 1.
</p>
<h2><span class="mw-headline" id="Cobordism_as_an_extraordinary_cohomology_theory">Cobordism as an extraordinary cohomology theory</span></h2>
<p>Every <a href="/enwiki/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> theory (real, complex etc.) has an <a href="/enwiki/wiki/Extraordinary_cohomology_theory" class="mw-redirect" title="Extraordinary cohomology theory">extraordinary cohomology theory</a> called <a href="/enwiki/wiki/K-theory" title="K-theory">K-theory</a>. Similarly, every cobordism theory Ω<sup><i>G</i></sup> has an <a href="/enwiki/wiki/Extraordinary_cohomology_theory" class="mw-redirect" title="Extraordinary cohomology theory">extraordinary cohomology theory</a>, with homology ("bordism") groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{n}^{G}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{n}^{G}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/545544d30268fa569595571c782a405ae86d91e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.991ex; height:3.009ex;" alt="\Omega _{n}^{G}(X)"/></span> and cohomology ("cobordism") groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{G}^{n}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{G}^{n}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ce5d419ab93416cfb3a9e72a1ba5189211bab736" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:6.991ex; height:3.009ex;" alt="\Omega _{G}^{n}(X)"/></span> for any space <i>X</i>. The generalized homology groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{*}^{G}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{*}^{G}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/5aae3e0e56095fb6461dfd4cc8a7585ec6528129" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.991ex; height:3.009ex;" alt="\Omega _{*}^{G}(X)"/></span> are <a href="/enwiki/wiki/Covariance" title="Covariance">covariant</a> in <i>X</i>, and the generalized cohomology groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{G}^{*}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{G}^{*}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3ae9b12c69cfb80acc1257d7c0c8a728def01d5b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:6.991ex; height:3.009ex;" alt="\Omega _{G}^{*}(X)"/></span> are <a href="/enwiki/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> in <i>X</i>. The cobordism groups defined above are, from this point of view, the homology groups of a point: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{n}^{G}=\Omega _{n}^{G}({\text{pt}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>pt</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{n}^{G}=\Omega _{n}^{G}({\text{pt}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4ffb307d3dfaf7979b1169416d49eb09d15a6cd3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.509ex; height:3.009ex;" alt="\Omega _{n}^{G}=\Omega _{n}^{G}({\text{pt}})"/></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{n}^{G}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{n}^{G}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/545544d30268fa569595571c782a405ae86d91e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.991ex; height:3.009ex;" alt="\Omega _{n}^{G}(X)"/></span> is the group of <i>bordism</i> classes of pairs (<i>M</i>, <i>f</i>) with <i>M</i> a closed <i>n</i>-dimensional manifold <i>M</i> (with G-structure) and <i>f</i> : <i>M</i> → <i>X</i> a map. Such pairs (<i>M</i>, <i>f</i>), (<i>N</i>, <i>g</i>) are <i>bordant</i> if there exists a G-cobordism (<i>W</i>; <i>M</i>, <i>N</i>) with a map <i>h</i> : <i>W</i> → <i>X</i>, which restricts to <i>f</i> on <i>M</i>, and to <i>g</i> on <i>N</i>.
</p><p>An <i>n</i>-dimensional manifold <i>M</i> has a <a href="/enwiki/wiki/Homology_(mathematics)" title="Homology (mathematics)">fundamental homology class</a> [<i>M</i>] ∈ <i>H<sub>n</sub></i>(<i>M</i>) (with coefficients in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/191bfe791873cb941d1bf8ee1d62434ea800f8a8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:3.875ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2}"/></span> in general, and in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="\mathbb {Z} "/></span> in the oriented case), defining a natural transformation
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\Omega _{n}^{G}(X)\to H_{n}(X)\\(M,f)\mapsto f_{*}[M]\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo stretchy="false">]</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\Omega _{n}^{G}(X)\to H_{n}(X)\\(M,f)\mapsto f_{*}[M]\end{cases}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bcfb60807c7960328b69d6b79cfa0f4dd0d8f007" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:20.039ex; height:6.176ex;" alt="{\begin{cases}\Omega _{n}^{G}(X)\to H_{n}(X)\\(M,f)\mapsto f_{*}[M]\end{cases}}"/></span></dd></dl>
<p>which is far from being an isomorphism in general.
</p><p>The bordism and cobordism theories of a space satisfy the <a href="/enwiki/wiki/Eilenberg%E2%80%93Steenrod_axioms" title="Eilenberg–Steenrod axioms">Eilenberg–Steenrod axioms</a> apart from the dimension axiom. This does not mean that the groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{G}^{n}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{G}^{n}(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ce5d419ab93416cfb3a9e72a1ba5189211bab736" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:6.991ex; height:3.009ex;" alt="\Omega _{G}^{n}(X)"/></span> can be effectively computed once one knows the cobordism theory of a point and the homology of the space <i>X</i>, though the <a href="/enwiki/wiki/Atiyah%E2%80%93Hirzebruch_spectral_sequence" title="Atiyah–Hirzebruch spectral sequence">Atiyah–Hirzebruch spectral sequence</a> gives a starting point for calculations. The computation is only easy if the particular cobordism theory <a href="#Cobordism_as_an_extraordinary_cohomology_theory">reduces to a product of ordinary homology theories</a>, in which case the bordism groups are the ordinary homology groups
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{n}^{G}(X)=\sum _{p+q=n}H_{p}(X;\Omega _{q}^{G}({\text{pt}})).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mi>n</mi>
</mrow>
</munder>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>;</mo>
<msubsup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>pt</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{n}^{G}(X)=\sum _{p+q=n}H_{p}(X;\Omega _{q}^{G}({\text{pt}})).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/fe72da722eb8ae4114b8af662ec4fc075a0c867e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:31.272ex; height:5.676ex;" alt="{\displaystyle \Omega _{n}^{G}(X)=\sum _{p+q=n}H_{p}(X;\Omega _{q}^{G}({\text{pt}})).}"/></span></dd></dl>
<p>This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably <a href="/enwiki/wiki/Pontrjagin%E2%80%93Thom_construction#Framed_cobordism" class="mw-redirect" title="Pontrjagin–Thom construction">framed cobordism</a>, oriented cobordism and <a href="/enwiki/wiki/Complex_cobordism" title="Complex cobordism">complex cobordism</a>. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the <a href="/enwiki/wiki/Homotopy_groups_of_spheres" title="Homotopy groups of spheres">homotopy groups of spheres</a>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup>
</p><p>Cobordism theories are represented by <a href="/enwiki/wiki/Thom_spectrum" class="mw-redirect" title="Thom spectrum">Thom spectra</a> <i>MG</i>: given a group <i>G</i>, the Thom spectrum is composed from the <a href="/enwiki/wiki/Thom_space" title="Thom space">Thom spaces</a> <i>MG<sub>n</sub></i> of the <a href="/enwiki/wiki/Tautological_bundle" title="Tautological bundle">standard vector bundles</a> over the <a href="/enwiki/wiki/Classifying_space" title="Classifying space">classifying spaces</a> <i>BG<sub>n</sub></i>. Note that even for similar groups, Thom spectra can be very different: <i>MSO</i> and <i>MO</i> are very different, reflecting the difference between oriented and unoriented cobordism.
</p><p>From the point of view of spectra, unoriented cobordism is a product of <a href="/enwiki/wiki/Eilenberg%E2%80%93MacLane_spectrum" class="mw-redirect" title="Eilenberg–MacLane spectrum">Eilenberg–MacLane spectra</a> – <i>MO</i> = <i>H</i>(<span class="texhtml mvar" style="font-style:italic;">π</span><sub>∗</sub>(<i>MO</i>)) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum <i>MSO</i> is rather more complicated than <i>MO</i>.
</p>
<h2><span class="mw-headline" id="See_also">See also</span></h2>
<ul><li><a href="/enwiki/wiki/H-cobordism" title="H-cobordism"><i>h</i>-cobordism</a></li>
<li><a href="/enwiki/wiki/Link_concordance" title="Link concordance">Link concordance</a></li>
<li><a href="/enwiki/wiki/List_of_cohomology_theories" title="List of cohomology theories">List of cohomology theories</a></li>
<li><a href="/enwiki/wiki/Symplectic_filling" title="Symplectic filling">Symplectic filling</a></li>
<li><a href="/enwiki/wiki/Cobordism_hypothesis" title="Cobordism hypothesis">Cobordism hypothesis</a></li>
<li><a href="/enwiki/wiki/Cobordism_ring" title="Cobordism ring">Cobordism ring</a></li>
<li><a href="/enwiki/wiki/Timeline_of_bordism" title="Timeline of bordism">Timeline of bordism</a></li></ul>
<h2><span class="mw-headline" id="Notes">Notes</span></h2>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The notation "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="(n+1)"/></span>-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional bordism" refers to a 5-dimensional bordism between 4-dimensional manifolds or a 6-dimensional bordism between 5-dimensional manifolds.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1133582631">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("/upwiki/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("/upwiki/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("/upwiki/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("/upwiki/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFStong1968" class="citation book cs1"><a href="/enwiki/wiki/Robert_Evert_Stong" title="Robert Evert Stong">Stong, Robert E.</a> (1968). <i>Notes on bordism theory</i>. Princeton, NJ: <a href="/enwiki/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+bordism+theory&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=1968&rft.aulast=Stong&rft.aufirst=Robert+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">While every cobordism is a cospan, the category of cobordisms is <i>not</i> a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that <i>M</i> and <i>N</i> form a partition of the boundary of <i>W</i> is a global constraint.</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFSwitzer2002" class="citation cs2">Switzer, Robert M. (2002), <i>Algebraic topology—homotopy and homology</i>, Classics in Mathematics, Berlin, New York: <a href="/enwiki/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-3-540-42750-6" title="Special:BookSources/978-3-540-42750-6"><bdi>978-3-540-42750-6</bdi></a>, <a href="/enwiki/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1886843">1886843</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+topology%E2%80%94homotopy+and+homology&rft.place=Berlin%2C+New+York&rft.series=Classics+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-3-540-42750-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1886843%23id-name%3DMR&rft.aulast=Switzer&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span>, chapter 12</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFRavenel1986" class="citation book cs1">Ravenel, D.C. (April 1986). <i>Complex cobordism and stable homotopy groups of spheres</i>. Academic Press. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-12-583430-6" title="Special:BookSources/0-12-583430-6"><bdi>0-12-583430-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+cobordism+and+stable+homotopy+groups+of+spheres&rft.pub=Academic+Press&rft.date=1986-04&rft.isbn=0-12-583430-6&rft.aulast=Ravenel&rft.aufirst=D.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></span>
</li>
</ol></div></div>
<h2><span class="mw-headline" id="References">References</span></h2>
<ul><li><a href="/enwiki/wiki/Frank_Adams" title="Frank Adams">John Frank Adams</a>, <i>Stable homotopy and generalised homology</i>, Univ. Chicago Press (1974).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFAnosovVoitsekhovskii2001" class="citation cs2"><a href="/enwiki/wiki/Dmitri_Anosov" title="Dmitri Anosov">Anosov, Dmitri V.</a>; Voitsekhovskii, M. I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=bordism">"bordism"</a>, <i><a href="/enwiki/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/enwiki/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=bordism&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Anosov&rft.aufirst=Dmitri+V.&rft.au=Voitsekhovskii%2C+M.+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dbordism&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><a href="/enwiki/wiki/Michael_Atiyah" title="Michael Atiyah">Michael F. Atiyah</a>, <i>Bordism and cobordism</i> Proc. Camb. Phil. Soc. 57, pp. 200–208 (1961).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFDieudonné1989" class="citation book cs1"><a href="/enwiki/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean Alexandre</a> (1989). <span class="cs1-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofalgebra0000dieu_g9a3"><i>A history of algebraic and differential topology, 1900–1960</i></a></span>. Boston: Birkhäuser. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-0-8176-3388-2" title="Special:BookSources/978-0-8176-3388-2"><bdi>978-0-8176-3388-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+algebraic+and+differential+topology%2C+1900%E2%80%931960&rft.place=Boston&rft.pub=Birkh%C3%A4user&rft.date=1989&rft.isbn=978-0-8176-3388-2&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean+Alexandre&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofalgebra0000dieu_g9a3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFKosinski2007" class="citation journal cs1">Kosinski, Antoni A. (October 19, 2007). "Differential Manifolds". Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Differential+Manifolds&rft.date=2007-10-19&rft.aulast=Kosinski&rft.aufirst=Antoni+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{<a href="/enwiki/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-hidden-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/enwiki/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFMadsenMilgram1979" class="citation book cs1"><a href="/enwiki/wiki/Ib_Madsen" title="Ib Madsen">Madsen, Ib</a>; Milgram, R. James (1979). <i>The classifying spaces for surgery and cobordism of manifolds</i>. <a href="/enwiki/wiki/Princeton,_New_Jersey" title="Princeton, New Jersey">Princeton, New Jersey</a>: <a href="/enwiki/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-0-691-08226-4" title="Special:BookSources/978-0-691-08226-4"><bdi>978-0-691-08226-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+classifying+spaces+for+surgery+and+cobordism+of+manifolds&rft.place=Princeton%2C+New+Jersey&rft.pub=Princeton+University+Press&rft.date=1979&rft.isbn=978-0-691-08226-4&rft.aulast=Madsen&rft.aufirst=Ib&rft.au=Milgram%2C+R.+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFMilnor1962" class="citation journal cs1"><a href="/enwiki/wiki/John_Milnor" title="John Milnor">Milnor, John</a> (1962). "A survey of cobordism theory". <i><a href="/enwiki/wiki/L%27Enseignement_Math%C3%A9matique" class="mw-redirect" title="L'Enseignement Mathématique">L'Enseignement Mathématique</a></i>. <b>8</b>: 16–23. <a href="/enwiki/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0013-8584">0013-8584</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=L%27Enseignement+Math%C3%A9matique&rft.atitle=A+survey+of+cobordism+theory&rft.volume=8&rft.pages=16-23&rft.date=1962&rft.issn=0013-8584&rft.aulast=Milnor&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><a href="/enwiki/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Sergei Novikov</a>, <i>Methods of algebraic topology from the point of view of cobordism theory</i>, Izv. Akad. Nauk SSSR Ser. Mat. <b>31</b> (1967), 855–951.</li>
<li><a href="/enwiki/wiki/Lev_Pontryagin" title="Lev Pontryagin">Lev Pontryagin</a>, <i>Smooth manifolds and their applications in homotopy theory</i> American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959).</li>
<li><a href="/enwiki/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a>, <i>On the formal group laws of unoriented and complex cobordism theory</i> Bull. Amer. Math. Soc., 75 (1969) pp. 1293–1298.</li>
<li><a href="/enwiki/wiki/Douglas_Ravenel" title="Douglas Ravenel">Douglas Ravenel</a>, <i>Complex cobordism and stable homotopy groups of spheres</i>, Acad. Press (1986).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFYuli_B._Rudyak2001" class="citation cs2">Yuli B. Rudyak (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Cobordism">"Cobordism"</a>, <i><a href="/enwiki/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/enwiki/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cobordism&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.au=Yuli+B.+Rudyak&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DCobordism&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><a href="/enwiki/wiki/Yuli_Rudyak" title="Yuli Rudyak">Yuli B. Rudyak</a>, <i>On Thom spectra, orientability, and (co)bordism</i>, Springer (2008).</li>
<li><a href="/enwiki/wiki/Robert_Evert_Stong" title="Robert Evert Stong">Robert E. Stong</a>, <i>Notes on cobordism theory</i>, Princeton Univ. Press (1968).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFTaimanov2007" class="citation book cs1">Taimanov, Iskander A. (2007). <i>Topological library. Part 1: cobordisms and their applications</i>. Series on Knots and Everything. Vol. 39. <a href="/enwiki/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">S. Novikov</a> (ed.). World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-981-270-559-4" title="Special:BookSources/978-981-270-559-4"><bdi>978-981-270-559-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+library.+Part+1%3A+cobordisms+and+their+applications&rft.series=Series+on+Knots+and+Everything&rft.pub=World+Scientific+Publishing+Co.+Pte.+Ltd.%2C+Hackensack%2C+NJ&rft.date=2007&rft.isbn=978-981-270-559-4&rft.aulast=Taimanov&rft.aufirst=Iskander+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li>
<li><a href="/enwiki/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a>, <i>Quelques propriétés globales des variétés différentiables</i>, <a href="/enwiki/wiki/Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Commentarii Mathematici Helvetici</a> 28, 17-86 (1954).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"/><cite id="CITEREFWall1960" class="citation journal cs1"><a href="/enwiki/wiki/C._T._C._Wall" title="C. T. C. Wall">Wall, C. T. C.</a> (1960). "Determination of the cobordism ring". <i><a href="/enwiki/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. Second Series. The Annals of Mathematics, Vol. 72, No. 2. <b>72</b> (2): 292–311. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970136">10.2307/1970136</a>. <a href="/enwiki/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="/enwiki/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970136">1970136</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Determination+of+the+cobordism+ring&rft.volume=72&rft.issue=2&rft.pages=292-311&rft.date=1960&rft.issn=0003-486X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970136%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1970136&rft.aulast=Wall&rft.aufirst=C.+T.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACobordism" class="Z3988"></span></li></ul>
<h2><span class="mw-headline" id="External_links">External links</span></h2>
<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110719102848/http://www.map.him.uni-bonn.de/Bordism">Bordism</a> on the Manifold Atlas.</li>
<li><a rel="nofollow" class="external text" href="https://archive.today/20120529222452/http://www.map.him.uni-bonn.de/B-Bordism">B-Bordism</a> on the Manifold Atlas.</li></ul>
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<ul><li><a href="/enwiki/wiki/General_topology" title="General topology">General (point-set)</a></li>
<li><a href="/enwiki/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li>
<li><a href="/enwiki/wiki/Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>
<li><a href="/enwiki/wiki/Continuum_(topology)" title="Continuum (topology)">Continuum</a></li>
<li><a href="/enwiki/wiki/Differential_topology" title="Differential topology">Differential</a></li>
<li><a href="/enwiki/wiki/Geometric_topology" title="Geometric topology">Geometric</a>
<ul><li><a href="/enwiki/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>
<li><a href="/enwiki/wiki/Homology_(mathematics)" title="Homology (mathematics)">Homology</a>
<ul><li><a href="/enwiki/wiki/Cohomology" title="Cohomology">cohomology</a></li></ul></li>
<li><a href="/enwiki/wiki/Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>
<li><a href="/enwiki/wiki/Digital_topology" title="Digital topology">Digital</a></li></ul>
</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><a href="/enwiki/wiki/Klein_bottle" title="Klein bottle"><img alt="Computer graphics rendering of a Klein bottle" src="/upwiki/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/60px-Kleinsche_Flasche.png" decoding="async" width="60" height="80" srcset="/upwiki/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/90px-Kleinsche_Flasche.png 1.5x, /upwiki/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/120px-Kleinsche_Flasche.png 2x" data-file-width="1171" data-file-height="1561" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Open_set" title="Open set">Open set</a> / <a href="/enwiki/wiki/Closed_set" title="Closed set">Closed set</a></li>
<li><a href="/enwiki/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li>
<li><a href="/enwiki/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>
<li><a href="/enwiki/wiki/Topological_space" title="Topological space">Space</a>
<ul><li><a href="/enwiki/wiki/Compact_space" title="Compact space">compact</a></li>
<li><a href="/enwiki/wiki/Connected_space" title="Connected space">Connected</a></li>
<li><a href="/enwiki/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>
<li><a href="/enwiki/wiki/Metric_space" title="Metric space">metric</a></li>
<li><a href="/enwiki/wiki/Uniform_space" title="Uniform space">uniform</a></li></ul></li>
<li><a href="/enwiki/wiki/Homotopy" title="Homotopy">Homotopy</a>
<ul><li><a href="/enwiki/wiki/Homotopy_group" title="Homotopy group">homotopy group</a></li>
<li><a href="/enwiki/wiki/Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>
<li><a href="/enwiki/wiki/Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>
<li><a href="/enwiki/wiki/CW_complex" title="CW complex">CW complex</a></li>
<li><a href="/enwiki/wiki/Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>
<li><a href="/enwiki/wiki/Manifold" title="Manifold">Manifold</a></li>
<li><a href="/enwiki/wiki/Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>
<li><a href="/enwiki/wiki/Second-countable_space" title="Second-countable space">Second-countable space</a></li>
<li><a class="mw-selflink selflink">Cobordism</a></li></ul>
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<ul><li><a href="/enwiki/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>
<li><a href="/enwiki/wiki/Betti_number" title="Betti number">Betti number</a></li>
<li><a href="/enwiki/wiki/Winding_number" title="Winding number">Winding number</a></li>
<li><a href="/enwiki/wiki/Chern_class" title="Chern class">Chern number</a></li>
<li><a href="/enwiki/wiki/Orientability" title="Orientability">Orientability</a></li></ul>
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<ul><li><a href="/enwiki/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>
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<li><a href="/enwiki/wiki/Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>
<li><a href="/enwiki/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>
<li><a href="/enwiki/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>
<li><a href="/enwiki/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>
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<li><img alt="" src="/upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" title="List-Class article" width="16" height="16" class="noviewer" srcset="/upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, /upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/enwiki/wiki/List_of_topology_topics" title="List of topology topics">Topics</a>
<ul><li><a href="/enwiki/wiki/List_of_general_topology_topics" title="List of general topology topics">general</a></li>
<li><a href="/enwiki/wiki/List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>
<li><a href="/enwiki/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>
<li><img alt="" src="/upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" title="List-Class article" width="16" height="16" class="noviewer" srcset="/upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, /upwiki/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/enwiki/wiki/List_of_important_publications_in_mathematics#Topology" title="List of important publications in mathematics">Publications</a></li></ul>
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