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Thom isomorphism

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cohomology

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Contents

Idea

For HH a multiplicative cohomology theory, and VXV \to X a vector bundle of rank nn, which is HH-orientable, the Thom isomorphism is the morphism

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

from the cohomology of XX to the cohomology of the Thom space Th(V)Th(V), induced by cup product with a Thom class uH n(Th(V))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

Concretely

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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

See also

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

Lecture notes include

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)

See also

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

  • Yuli 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

Revised on February 18, 2016 14:36:05 by Urs Schreiber (195.37.209.180)