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