group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
from the cohomology of $X$ to the reduced 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.
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A fully general abstract discussion is around page 30, 31 of (ABGHR).
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The original proof that the Thom isomorphism is indeed an isomorphism is due to
Textbook accounts include
See also
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)
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
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