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 morphism
from the cohomology of to the cohomology of the Thom space , induced by cup product with a Thom class . 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.
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 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
- W. Cockcroft, On the Thom isomorphism Theorem, Mathematical Proceedings of the Cambridge Philosophical Society (1962), 58 (pdf)
Lecture notes include
John Francis, Topology of manifolds, course notes (2010) (web), Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)
Akhil Mathew, The Thom isomorphism theorem, 2010
Johannes Ebert, sections 2.3, 2.4 of A lecture course on Cobordism Theory, 2012 (pdf)
Dan Freed, lecture 8 of Bordism: old and new, 2013 (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)
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