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Concept
Thom space
Summary
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
One way to construct this space is as follows. Let
be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber is an -dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let be the unit ball bundle with respect to our orthogonal structure, and let be the unit sphere bundle, then the Thom space is the quotient of topological spaces. is a pointed space with the image of in the quotient as basepoint. If B is compact, then is the one-point compactification of E.
For example, if E is the trivial bundle , then and . Writing for B with a disjoint basepoint, is the smash product of and ; that is, the n-th reduced suspension of .
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 coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
Let be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism
for all k greater than or equal to 0, where the right hand side is reduced cohomology.
This theorem was formulated and proved by René Thom in his famous 1952 thesis.
We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of , B with a disjoint point added (cf. #Construction of the Thom space.) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:
Let be a ring and be an oriented real vector bundle of rank n.
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