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.**) (more…)