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 .
Note that lies necessarily in .
If a Thom class for exists, is said to be orientable with respect to the cohomology theory .
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 .
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.
|Date of creation||2013-03-22 15:40:48|
|Last modified on||2013-03-22 15:40:48|
|Last modified by||antonio (1116)|
|Defines||orientability with respect to a generalized homology theory|