The Thom space of a vector bundle over a topological space is the topological space obtained by first forming the disk bundle of (unit) disks in the fibers of and then identifying to a point the boundaries of all the disks, i.e. forming the quotient by the sphere bundle :
(N.B.: this is a quotient of the total spaces of the bundles taken in , not a bundle quotient in .) This is equivalently the mapping cone
in Top of the sphere bundle of . Therefore more generally, for any -bundle over , its Thom space is the the mapping cone
of the bundle projection.
For a compact topological space, the is a model for the one-point compactification of the total space .
The Thom space of the universal rank- vector bundle over the classifying space of the orthogonal group is usuelly denoted . As ranges, these spaces form the Thom spectrum.
The Thom space of the rank-0 bundle over is the space with a basepoint freely adjoined:
For a vector bundle and its fiberwise direct sum with the trivial rank vector bundle we have
is the smash product of the Thom space of with the -sphere (the -fold suspension).
In particular, if is a trivial vector bundle of rank , then
is the smash product of the -sphere with with one base point freely adjoined (the -fold suspension).
The Thom isomorphism for Thom spaces was originally found in
- René Thom, Quelques propriétés globales des variétés différentiables Comm. Math. Helv. , 28 (1954) pp. 17–86
For general discussion see
Michael Atiyah, Thom complexes, Proc. London Math. Soc. 11 (1961) pp. 291–310
Yuli Rudyak, On Thom spectra, orientability, and cobordism, Springer 1998 googB
eom, Thom space
Dale Husemöller, Fibre bundles , McGraw-Hill (1966)
myyn.org Thom space, Thom class, Thom isomorphism theorem
Robert Stong, Notes on cobordism theory , Princeton Univ. Press (1968)
W.B. Browder, Surgery on simply-connected manifolds , Springer (1972)
Revised on February 10, 2016 06:08:29
by Todd Trimble