Thom Space - The Thom Class

The Thom Class

The isomorphism of the theorem is explicitly known: there is a certain cohomology class, the Thom class, in the kth cohomology group of the Thom space. Denote this Thom class by U. Then for a class b in the cohomology of the base, we can compute the Thom isomorphism via the pullback of the bundle projection and the cohomology cup product:

In particular, the Thom isomorphism sends the identity element of H*(B) to U.