nLab
Thom isomorphism

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

$$(-)\cdot u:H^{•}(X)\stackrel{\simeq }{\to }{H}^{•+n}(\mathrm{Th}(V)).$$(-) \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 $\mathrm{Th}(V)$, induced by multiplication with a Thom class $u\in H^n(\mathrm{Th}(V))$.

We think of this from left to right as wedging 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.

Related concepts

• Thom space, Thom spectrum

• fiber integration

• Pontrjagin-Thom collapse map

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

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology

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

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

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)

See also

• A. Dold, Relations between ordinary and extraordinary homology , Colloq. Algebraic Topology, August 1–10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2–9

• Yu.B. Rudyak, On the Thom–Dold isomorphism for nonorientable bundles Soviet Math. Dokl. , 22 (1980) pp. 842–844 Dokl. Akad. Nauk. SSSR , 255 : 6 (1980) pp. 1323–1325

• R.M. Switzer, Algebraic topology - homotopy and homology , Springer (1975)

• myyn.org (Planetmath) Thom space, Thom class, Thom isomorphism theorem