Thom class
Let be a generalized cohomology theory (for example, let , singular cohomology with integer coefficients). Let be a vector bundle of dimension over a topological space . Assume for convenience that has a Riemannian metric, so that we may speak of its associated sphere and disk bundles, and respectively.
Let , and consider the fibers and . Since is homeomorphic to the -sphere, the Eilenberg-Steenrod axioms for imply that is isomorphic to the coefficient group of . In fact, is a free module of rank one over the ring .
Definition 1
An element is said to be a Thom class for if, for every , the restriction of to is an -module generator.
Note that lies necessarily in .
Definition 2
If a Thom class for exists, is said to be orientable with respect to the cohomology theory .
Remark 1
Notice that we may consider as an element of the reduced -cohomology group , where is the Thom space of . 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 . For example, the pair (where is included in as the zero section) is homotopy equivalent to .
Remark 2
If is singular cohomology with integer coefficients, then 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 |