An Introduction to Cobordism Theory

The notion of cobordism is simple; two manifolds M and N are said to be cobordant if their disjoint union is the boundary of some other manifold. Given the extreme difficulty of the classification of manifolds it would seem very unlikely that much progress could be made in classifying manifolds up to cobordism. However, René Thom, in his remarkable, if unreadable, 1954 paper Quelques propriétés globales des variétés differentiables [22], gave the full solution to this problem for unoriented manifolds, as well as many powerful insights into the methods for solving it in the cases of manifolds with additional structure. It was largely for this work that Thom was awarded the Fields medal in 1958. The key step was the reduction of the cobordism problem to a homotopy problem, although the homotopy problem is still far from trivial. This was later generalized by Lev Pontrjagin, and this result is now known as the Thom-Pontrjagin theorem. The first part of this paper will work towards the proof of the generalized Thom-Pontrjagin theorem. We will begin by abstracting the usual notion of cobordism on manifolds. Personally distasteful as this may be, it will be useful to have an abstract definition phrased in the language of category theory. We will then return to the specific situation of manifolds. The notion of a (B, f) structure on a manifold will be introduced, primarily as a means of unifying the many different special structures (orientations, complex structures, spin structures, etc.) which can be put on a manifold. The class of (B, f) manifolds can be made in a natural way into a cobordism category, and in this way we will define the cobordism groups Ω(B, f). Once these ideas are all established, we will turn towards the