Thom space

Let ξX be a vector bundleMathworldPlanetmath over a topological spaceMathworldPlanetmath X. Assume that ξ has a Riemannian metric. We can form its associated disk bundle D(ξ) and its associated sphere bundle S(ξ), by letting


The Thom space of ξ is defined to be the quotient spaceMathworldPlanetmath D(ξ)/S(ξ), obtained by taking the disk bundle and collapsing the sphere bundle to a point. Notice that this makes the Thom space naturally into a based topological spacePlanetmathPlanetmath.

Two common forms of notation for the Thom space are Th(ξ) and Xξ.

Remark 1

If ξ=X×d is a trivial vector bundle, then its Thom space is homeomorphicMathworldPlanetmath to ΣdX+, where X+ stands for X with an added disjoint basepoint, and Σd stands for the based suspension iterated d times. Thus, we may think of Xξ as a “twisted suspension” of X+.

Remark 2

If X is compactPlanetmathPlanetmath, then Xξ is homeomorphic as a based space to the one-point compactification of ξ. Even if X is not compact, Xξ can be obtained by doing a one-point compactification on each fiber and then collapsing the resulting sectionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of points at infinity to a point.

Remark 3

The choice of Riemannian metric on ξ does not change the homeomorphismMathworldPlanetmath type of Xξ, and, by the previous remark, the Thom space can be described without reference to associated disk and sphere bundles.

Title Thom space
Canonical name ThomSpace
Date of creation 2013-03-22 15:40:46
Last modified on 2013-03-22 15:40:46
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 5
Author antonio (1116)
Entry type Definition
Classification msc 57R22
Classification msc 55R25
Defines Thom space
Defines disk bundle
Defines sphere bundle