Manifolds and cobordisms
Special and general types
For a multiplicative cohomology theory, and a vector bundle of rank , which is -orientable, the Thom isomorphism is the isomorphism
from the cohomology of to the cohomology of the Thom space , induced by multiplication with a Thom class .
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 Thom isomorphism is used to define fiber integration of multiplicative cohomology theories.
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
- Martin Jakob, A note on the Thom isomorphism in geometric (co)homology (arXiv:math/0403540)
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
Revised on July 5, 2013 19:08:11
by Urs Schreiber