Today I would like to take a break from the index theorem, and blog about a result of Wu, that the Stiefel-Whitney classes of a compact manifold (i.e. those of the tangent bundle) are homotopy invariant. It is not even a priori obvious that the Stiefel-Whitney classes are homeomorphism invariant; note that “homeomorphic” is a strictly weaker relation than “diffeomorphic” for compact manifolds, a result first due to Milnor. But in fact the argument shows even that the Stiefel-Whitney classes (of the tangent bundle) can be worked out solely in terms of the structure of the cohomology ring as a module over the Steenrod algebra.

Here is the idea. When A \subset M is a closed submanifold of a manifold, there is a lower shriek (Gysin) homomorphism from the cohomology of A to that of M; this is Poincaré dual to the restriction map in the other direction. We will see that the “fundamental class” of A (that is,  the image of 1 under this lower shriek map) corresponds to the mod 2 Euler (or top Stiefel-Whitney) class of the normal bundle. In the case of M \subset M \times M, the corresponding normal bundle is just the tangent bundle of M. But by other means we’ll be able to work out the Gysin map easily. Once we have this, the Steenrod operations determine the rest of the Stiefel-Whitney classes.