Thom class

Let h* be a generalized cohomology theory (for example, let h*=H*, singular cohomology with integer coefficients). Let ξX be a vector bundleMathworldPlanetmath of dimensionMathworldPlanetmath d over a topological spaceMathworldPlanetmath X. Assume for convenience that ξ has a Riemannian metric, so that we may speak of its associated sphere and disk bundles, S(ξ) and D(ξ) respectively.

Let xX, and consider the fibers S(ξx) and D(ξx). Since D(ξx)/S(ξx) is homeomorphicMathworldPlanetmath to the d-sphere, the Eilenberg-Steenrod axioms for h* imply that h*+d(D(ξx),S(ξx)) is isomorphic to the coefficient group h*(pt) of h*. In fact, h*(D(ξx),S(ξx)) is a free moduleMathworldPlanetmathPlanetmath of rank one over the ring h*(pt).

Definition 1

An element τh*(D(ξ),S(ξ)) is said to be a Thom class for ξ if, for every xX, the restrictionPlanetmathPlanetmath of τ to h*(D(ξx),S(ξx)) is an h*(pt)-module generatorPlanetmathPlanetmathPlanetmath.

Note that τ lies necessarily in hd(D(ξ),S(ξ)).

Definition 2

If a Thom class for ξ exists, ξ is said to be orientable with respect to the cohomology theory h*.

Remark 1

Notice that we may consider τ as an element of the reduced h*-cohomology groupPlanetmathPlanetmath h~*(Xξ), where Xξ is the Thom space D(ξ)/S(ξ) of ξ. As is the case in the definition of the Thom space, the Thom class may be defined without reference to associated disk and sphere bundles, and hence to a Riemannian metric on ξ. For example, the pair (ξ,ξ-X) (where X is included in ξ as the zero sectionMathworldPlanetmath) is homotopy equivalent to (D(ξ),S(ξ)).

Remark 2

If h* is singular cohomology with integer coefficients, then ξ has a Thom class if and only if it is an orientable vector bundle in the ordinary sense, and the choices of Thom class are in one-to-one correspondence with the orientations.

Title Thom class
Canonical name ThomClass
Date of creation 2013-03-22 15:40:48
Last modified on 2013-03-22 15:40:48
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 5
Author antonio (1116)
Entry type Definition
Classification msc 55-00
Related topic Orientation2
Defines orientability with respect to a generalized homology theory