Classes of bundles
Examples and Applications
The Thom space of a real vector bundle over a topological space is the topological space obtained by first forming the disk bundle of (unit) disks in the fibers of (with respect to a metric given by any choice of orthogonal structure) and then identifying to a point the boundaries of all the disks, i.e. forming the quotient topological space 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 n-sphere-fiber bundle over (spherical fibration), its Thom space is the the mapping cone
of the bundle projection.
For a compact topological space, is a model for the one-point compactification of the total space .
The Thom space of the rank- universal vector bundle over the classifying space of the orthogonal group is usuelly denoted . As ranges, these spaces form the Thom spectrum.
Let be a topological space and let be a vector bundle over of rank , which is associated to an O(n)-principal bundle. Equivalently this means that is the pullback of the universal vector bundle over the classifying space. Since preserves the metric on , by definition, such inherits the structure of a metric space-fiber bundle. With respect to this structure:
the unit disk bundle is the subbundle of elements of norm ;
the unit sphere bundle is the subbundle of elements of norm ;
the Thom space is the cofiber (formed in Top (prop.)) of
canonically regarded as a pointed topological space.
If is a general real vector bundle, then there exists an isomorphism to an -associated bundle and the Thom space of is, up to based homeomorphism, that of this orthogonal bundle.
Behaviour under direct sum of vector bundles
Let be two real vector bundles. Then the Thom space (def. 1) of the direct sum of vector bundles is expressed in terms of the Thom space of the pullbacks and of to the disk/sphere bundle of as
(Since a point at radius in is a point of radius in and a point of radius in .)
Apply prop. 1 with and . Since is a trivial bundle, then
(as a bundle over ) and similarly
By prop. 2 and remark 1 the Thom space (def. 1) of a trivial vector bundle of rank is the -fold suspension of the base space
Therefore a general Thom space may be thought of as “twisted suspension”, with twist encoded by a vector bundle (or rather by its underlying spherical fibration). See at Thom spectrum – For infinity-module bundles for more on this.
For and to vector bundles, let be the direct sum of vector bundles of their pullbacks to . The corresponding Thom space is the smash product of the individual Thom spaces:
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)