Cobordism: Difference between revisions
JF Manning (talk | contribs) Added a picture of pants, and the definition of null-cobordant. |
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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. |
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 case, we say that ''M'' and ''N'' are '''cobordant'''. In this way we get an [[equivalence relation]] (cobordance) on manifolds. |
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For example, if ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a |
For example, if ''M'' consists of a [[circle]], and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a [[pair of pants]] ''L'' (see the figure at right). |
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Thus the pair of pants is a cobordism between ''M'' and ''N''. |
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[[Image:Pants.png|thumb|right| A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).]] |
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An ''n''-manifold ''M'' is said to be '''null-cobordant''' if there is a cobordism between ''M'' and the empty manifold; in other words, ''M'' is the entire boundary of some ''(n+1)''-manifold. For example, the circle is null-cobordant since it bounds a disk. |
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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. |
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. |
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Revision as of 04:47, 22 February 2006
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 case, we say that M and N are cobordant. In this way we get an equivalence relation (cobordance) 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 pair of pants L (see the figure at right). Thus the pair of pants is a cobordism between M and N.
An n-manifold M is said to be null-cobordant if there is a cobordism between M and the empty manifold; in other words, M is the entire boundary of some (n+1)-manifold. For example, the circle is null-cobordant since it bounds a disk.
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 (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.