Classification of manifolds up to diffeomorphism is a hard problem (deciding whether two given 4-manifolds are diffeomorphic is at least as hard as the word problem for groups). Thom's brilliant idea (which had been considered, before him, by Poincaré and Pontryagin) was to introduce a weaker equivalence relation than diffeomorphism: the equivalence relation of cobordism.

Two -manifolds are (unoriented) cobordant if their disjoint union is the boundary of an -manifold with boundary. We can see that we can compose cobordisms end to end, and every manifold is cobordant to itself as we have , so it is easy to check it is indeed an equivalence relation. In each dimension we can form an abelian group, with operation disjoint union, the -th cobordism group.

There are other types of cobordism, that arise by putting structure on our manifolds and requiring that the cobordisms preserve this, for example oriented cobordism: we only study oriented manifolds, and the oriented boundary of the manifold with boundary has to give the correct orientation on the -manifolds.

Pontryagin's success was to realise that certain cobordism groups end up being equal to stable homotopy groups of spheres, helping the calculation of the latter.

Thom largely reversed the process and generalised Pontryagin's idea, making an explicit correspondence between cobordism groups and homotopy groups of certain spaces, now called Thom spaces.

Another useful way to study cobordism is through characteristic classes. Many cobordism theories have particular characteristic classes associated to them, and it turns out that two manifolds are cobordant if and only if all the characteristic numbers of these two manifolds agree. This lead to great insights into the theory of characteristic classes, for example the idea of a generalised genus (a topological invariant) as a homomorphism from the cobordism ring to some other ring, for example the rationals. This is then of great interest in classifying manifolds, and allowed things such as Milnor's construction of exotic -spheres.

If you're already impatient to hear more, then you should come to MS.05 this thursday at 7:30 pm; you'll hear about all this and then much more (such as topological quantum field theories!).

After which we realise that we are cobordant to our future selves that are in the pub.