The Impact of Thom’s Cobordism Theory

At the 1958 International Congress of Mathematicians in Edinburgh, René Thom received one of two Field Medals for his development of cobordism. In his citation [11] Heinz Hopf described the definition of cobordism as one of those elementary and apparently trivial constructions which can hardly be expected to yield significant results. He compares this with Hurewicz’s definition of homotopy groups, a very simple idea which has turned out to be extremely fruitful. Hopf then points out that there is in fact a close link between cobordism and homotopy which Thom exploits. In fact the basic idea linking homotopy theory to differentiable manifolds goes back to a construction of Pontrjagin [13]. Given a smooth map f : Y → X between two compact, connected and oriented differentiable manifolds, the inverse image f−1 (p) of a regular value p ∈ X is an oriented submanifold F of Y with dimF = dimY −dimX. It is easy to see that the homology class of F in Y is independent of p and is a homotopy invariant of f : in fact it is the Poincaré dual of the cohomology class f∗ (u) where u is the fundamental class of X in top dimension. But the geometry of F contains more information about f than just this homology class. For example, when Y = S, X = S are spheres and f is the Hopf fibration, then F is a circle and any two such circles have linking number 1. Applied to any map f : S → S, we get in this way a homotopy invariant known as the Hopf invariant. In another direction the homological equivalence between two fibres Fp and Fq (for p, q ∈ X) can be strengthened to a more precise geometrical relation, namely that there is an oriented manifold W with boundary Fp and −Fq (i.e. Fq with the opposite orientation). This manifold W appears naturally as a submanifold of Y × I, where I is the unit interval defined by a generic path in X from p to q. In Thom’s terminology W is a cobordism between Fp and Fq. Pontrjagin’s idea was to use the geometry of F to deduce information about the homotopy of f , in the spirit of the Hopf invariant. In the early days of homotopy theory this geometric approach paid some dividends, but it was delicate to use (and could lead to mistakes). But in the early fifties powerful new algebraic methods were introduced into homotopy theory, notably the Leray-Serre spectral sequence for fibrations, and Pontrjagin’s method then became obsolete, but Thom turned the tables and used homotopy theory to attack the geometry of manifolds. Specifically he showed [15] that the abstractly defined cobordism groups could