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 is a closed submanifold of a manifold, there is a lower shriek (Gysin) homomorphism from the cohomology of to that of ; this is Poincaré dual to the restriction map in the other direction. We will see that the “fundamental class” of (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 , the corresponding normal bundle is just the tangent bundle of . 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.