Thom isomorphism theorem

Let ξX be a d-dimensional vector bundleMathworldPlanetmath over a topological spaceMathworldPlanetmath X, and let h* be a multiplicative generalized cohomology theory, such as ordinary cohomology. Let τhd(D(ξ),S(ξ)) be a Thom class for ξ, where D(ξ) and S(ξ) are the associated disk and sphere bundles of ξ.

Since h* is a multiplicative theory, there is a generalized cup productMathworldPlanetmath map


where the tensor productPlanetmathPlanetmath is over the coefficient ring h*(pt) of the theory. Using the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath p*:h*(X)h*(D(ξ)) induced by the homotopy equivalenceMathworldPlanetmathPlanetmath p:D(ξ)X, we obtain a homomorphismMathworldPlanetmathPlanetmath


taking α to p*(α)τ. Here Xξ stands for the Thom space D(ξ)/S(ξ) of ξ.

Thom isomorphism theoremT is an isomorphism h*(X)h~*+d(Xξ) of graded modules over h*(pt).

Remark 1

When ξ is a trivial bundleMathworldPlanetmath of dimensionMathworldPlanetmathPlanetmath 1, this generalizes the suspension isomorphism. In fact, a typical proof of this theorem for compactPlanetmathPlanetmath X proceeds by induction over the number of open sets in a trivialization of ξ, using the suspension isomorphism as the base case and the Mayer-Vietoris sequence to carry out the inductive step.

Remark 2

There is also a homologyMathworldPlanetmathPlanetmath Thom isomorphism h~*+d(Xξ)h*(X), in which the map is given by cap product with the Thom class rather than cup product.

Title Thom isomorphism theorem
Canonical name ThomIsomorphismTheorem
Date of creation 2013-03-22 15:40:52
Last modified on 2013-03-22 15:40:52
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 6
Author antonio (1116)
Entry type Theorem
Classification msc 55-00