# nLab Thom isomorphism

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

cohomology

# 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 morphism

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

from the cohomology of $X$ to the cohomology of the Thom space $Th(V)$, induced by cup product with a Thom class $u \in H^n(Th(V))$. That this is indeed an isomorphism follows from the nature of Thom classes, for instance via the Leray-Hirsch theorem (see e.g. Ebert 12, 2.3,2.4) or from running a Serre spectral sequence (e.g. Kochmann 96, section 2.6).

One may think of the Thom isomorphism from left to right as cupping with a generalized volume form on the fibers, and from right to left as performing fiber integration.

## Definition

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### General abstractly

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 proof that the Thom isomorphism is indeed an isomorphism is due to

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

Textbook accounts include

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

Lecture notes include

• John Francis, Topology of manifolds, course notes (2010) (web), Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)

• Johannes Ebert, sections 2.3, 2.4 of A lecture course on Cobordism Theory, 2012 (pdf)

• Dan Freed, lecture 8 of Bordism: old and new, 2013 (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)