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'[[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''&nbsp;+&nbsp;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&nbsp;''M''. Every closed manifold ''M'' is the boundary of the non-compact manifold ''M''&nbsp;×&nbsp;[0,&nbsp;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''&nbsp;=&nbsp;∂''W''₁ and ''N''&nbsp;=&nbsp;∂''W''₂, 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''&nbsp;+&nbsp;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''&nbsp;+&nbsp;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''&nbsp;+&nbsp;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''&nbsp;+&nbsp;1, then the level-set ''N'' := ''f''⁻¹(''c''&nbsp;+&nbsp;ε) is obtained from ''M'' := ''f''⁻¹(''c''&nbsp;−&nbsp;ε) by a ''p''-surgery. The inverse image ''W'' := ''f''⁻¹([''c''&nbsp;−&nbsp;ε, ''c''&nbsp;+&nbsp;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,&thinsp;1] such that ''f''⁻¹(0) = ''M'', ''f''⁻¹(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,&thinsp;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''&thinsp;′; ''N'', ''P'') is defined by gluing the right end of the first to the left end of the second, yielding (''W''&thinsp;′ ∪_''N'' ''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''&nbsp;+&nbsp;''k'') → ''BO''(''k''). Given a collection of spaces and maps ''X_k'' → ''X_k''₊₁ with maps ''X_k'' → ''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_k'' 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''&nbsp;×&nbsp;''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''₁, ..., ''i_k'' ≥ 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^''G'' 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_n''(''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_n'' of the [[tautological bundle|standard vector bundles]] over the [[classifying space]]s ''BG_n''. 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}}_∗(''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.&nbsp;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.&nbsp;1–114 (1959). *[[Daniel Quillen]], ''On the formal group laws of unoriented and complex cobordism theory'' Bull. Amer. Math. Soc., 75 (1969) pp.&nbsp;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]]'