The Thom Isomorphism
The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles. (We have stated the result in terms of Z2 coefficients to avoid complications arising from orientability.)
Let B, E, and p be as above. Then there is an isomorphism, now called a Thom isomorphism
for all i greater than or equal to 0, where the right hand side is reduced cohomology.
We can loosely interpret the theorem as being a generalization of the suspension isomorphism on (co)homology, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of B+, B with a disjoint point added.
This theorem was formulated and proved by René Thom in his 1952 thesis.
Read more about this topic: Thom Space