Thom’s construction of the Stiefel-Whitney classes

I’ve been trying to fix the (many) gaps in my knowledge of classical algebraic topology as of late, and will probably do a few posts in the near future on vector bundles, K-theory, and characteristic classes.

Let be a base space, and let be a real vector bundle. There are numerous constructions for the characteristic classes of . Recall that these are elements in the cohomology ring (for some ring) that measure, in some sense, the twisting or nontriviality of the bundle .

Over a smooth manifold , with a smooth vector bundle, a construction can be made in de Rham cohomology. Namely, one chooses a connection on , computes the curvature tensor of (which is an -valued 2-form on ), and then applies a suitable polynomial from matrices to polynomials to the curvature . One can show that this gives closed forms, whose de Rham cohomology class does not depend on the choice of connection. This is the subject of Chern-Weil theory, and it applies more generally to principal -bundles on a manifold for a Lie group.

But there is something that this approach misses: torsion. By working with de Rham cohomology (or equivalently, cohomology with -coefficients), the very interesting torsion phenomena that algebraic topologists care about is lost. For the purposes of this post, we’re interested in cohomology classes where the ground ring is , and so de Rham cohomology is out. However, in return, we have cohomology operations. We can use them instead.

 

The Thom isomorphism theorem

Suppose is a topological space, and is a vector bundle over . Here is a “reasonable” space, for instance a manifold or a CW complex. There are many familiar examples of how such bundles arise “in nature,” for instance the tangent bundle to a manifold. It is of interest to understand how the apparatus of algebraic topology applies to the bundle.

Now, clearly there is a deformation retraction of given by that deformation retracts to the image of the zero section. In particular, is homotopy equivalent to . So by itself might not all be that interesting.

Nonetheless, let us suppose that is a normed bundle, e.g. the tangent bundle to a manifold with a Riemannian metric. In this case, we can consider the set of elements of norm one; it is no longer a vector bundle over , but it is a fiber bundle (with fibers various spheres). This is homotopically very different from in general.

For a general bundle , we can still consider the subset consisting of nonzero elements. Then there is a map whose fibers are of the form . It is easy to see that if is a normed bundle, then deformation retracts onto the associated sphere bundle. So we might as well see what is like. By using the long exact sequence, we might try studying the pair .

Today, I want to talk about the cohomology of the pair . Namely, the main result is:

Theorem 1 (Thom) For all , there is an isomorphism (to be described)

if has dimension .

(To ease notation, all cohomology will henceforth be with -coefficients.)