nLab Thom isomorphism

Context

Manifolds and cobordisms

manifolds and cobordisms

Contents

Idea

For $H$ a multiplicative cohomology theory, and $V \to X$ a vector bundle of rank $n$, which is $H$-orientable, the Thom isomorphism is the isomorphism

$(-) \cdot u : H^\bullet(X) \stackrel{\simeq}{\to} H^{\bullet + n}(Th(V)) \,.$

from the cohomology of $X$ to the cohomology of the Thom space $Th(V)$, induced by multiplication with a Thom class $u \in H^n(Th(V))$.

We think of this from left to right as wedge-ing with a generalized volume form on the fibers, and from right to left as performing fiber integration.

Definition

A fully general abstract discussion is around page 30,31 of (ABGHR).

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Applications

• The Thom isomorphism is used to define fiber integration of multiplicative cohomology theories.

References

The original reference is

• René Thom, Quelques propriétés globales des variétés différentiables Comm. Math. Helv. , 28 (1954) pp. 17–86

• W. Cockcroft, On the Thom isomorphism Theorem Mathematical Proceedings of the Cambridge Philosophical Society (1962), 58 (pdf)

A review is in

• John Francis, Topology of manifolds, course notes (2010) (web)

Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)

A discussion in differential geometry with fiberwise compactly supported differential forms is around theorem 6.17 of

A comprehensive general abstract account for multiplicative cohomology theories in terms of E-infinity ring spectra is in

An alternative simple formulation in terms of geometric cycles as in bivariant cohomology theory is in

• Martin Jakob, A note on the Thom isomorphism in geometric (co)homology (arXiv:math/0403540)