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In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. We can say that two manifolds M and N are cobordant if their union is the complete boundary of a third manifold L; L is then called a cobordism between M and N. In this way we get an equivalence relation on manifolds.

For example, if M consists of a circle, and N of two circles, M and N together make up the boundary of a T-shaped tube manifold L homeomorphic to what is known as the pair of pants. (Here L can actually be taken as connected; since M is already a boundary of a disk, we could also say, less graphically, that M is cobordant to the empty manifold.)

The general bordism problem is to calculate the cobordism classes of suitable, more precisely formulated cobordism relations. We should, for example, mention the orientation question: assume all manifolds are smooth and oriented. Then the correct definition is in terms of M and ${\bar {N}}$ (reversed orientation) making up the boundary of L, with the induced orientations.

History

Bordism was explicitly introduced by Pontryagin in geometric work on manifolds. It came to prominence when 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 the extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s, in particular in the Hirzebruch Riemann-Roch theorem, and in the first proofs of the Atiyah-Singer index theorem.

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 In [[mathematics]], '''cobordism''' is a relation between [[manifold]]s, based on the idea of [[boundary]]. We can say that two manifolds ''M'' and ''N'' are '''cobordant''' if their union is the complete boundary of a third manifold ''L;'' ''L'' is then called a cobordism between ''M'' and ''N''.  In this way we get an [[equivalence relation]] on manifolds.
-For example, if ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a T-[[shape]]d tube manifold ''L''. (Here ''L'' can actually be taken as connected; since ''M'' is already a boundary of a [[disk (mathematics)|disk]], we could also say, less graphically, that ''M'' is cobordant to the empty manifold.)
+For example, if ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a T-[[shape]]d tube manifold ''L'' homeomorphic to what is known as the [[pair of pants]]. (Here ''L'' can actually be taken as connected; since ''M'' is already a boundary of a [[disk (mathematics)|disk]], we could also say, less graphically, that ''M'' is cobordant to the empty manifold.)
 The general '''bordism''' problem is to calculate the cobordism classes of suitable, more precisely formulated cobordism relations. We should, for example, mention the [[orientation (manifold)|orientation]] question: assume all manifolds are smooth and oriented. Then the correct definition is in terms of ''M'' and <math> \bar N</math> (reversed orientation) making up the boundary of ''L'', with the induced orientations.
 ==History==
+Bordism was explicitly introduced by [[Pontryagin]] in geometric work on manifolds. It came to prominence when [[Rene Thom|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 the [[extraordinary cohomology theory]], alongside [[K-theory]]. It performed an important role, historically speaking, in developments in topology in the 1950s, in particular in the [[Hirzebruch]] [[Riemann-Roch theorem]], and in the first proofs of the [[Atiyah-Singer index theorem]].
-Bordism was explicitly introduced by [[Pontryagin]] in geometric work on manifolds. It came to prominence when [[Rene Thom|Thom]] showed that cobordism groups could be computed by means of [[homotopy theory]] (the [[Thom complex]] construction). Cobordism theory became part of the apparatus of the [[extraordinary cohomology theory]], alongside [[K-theory]]. It performed an important role, historically speaking, in developments in topology in the 1950s, in particular in the [[Hirzebruch]] [[Riemann-Roch theorem]], and in the first proofs of the [[Atiyah-Singer index theorem]].
 ==See also==