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.
Read more about this topic: Thom Space
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“Ours is the old, old story of every uprising race or class or order. The work of elevation must be wrought by ourselves or not at all.”
—Frances Power Cobbe (1822–1904)