Cobordism
|
|
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. (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 <math> \bar N<math> (reversed orientation) making up the boundary of L, with the induced orientations.
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 (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.
A cobordism W between M and N is an h-cobordism if the inclusion maps
- M → W
and
- N → W
are homotopy equivalences. The h-cobordism theorem states that if W is a compact smooth h-cobordism between M and N, and if in addition M and N are simply connected and of dimension > 4, then W is diffeomorphic to M × [0, 1] and M is diffeomorphic to N. If the assumption that M and N are simply connected is dropped, the theorem becomes false. It is true, however, if (and only if) the Whitehead torsion τ(W, M) vanishes; this is the s-cobordism theorem.ru:Бордизм