Thom isomorphism theorem
Let be a -dimensional vector bundle over a topological space , and let be a multiplicative generalized cohomology theory, such as ordinary cohomology. Let be a Thom class for , where and are the associated disk and sphere bundles of .
Since is a multiplicative theory, there is a generalized cup product map
taking to . Here stands for the Thom space of .
Thom isomorphism theorem is an isomorphism of graded modules over .
When is a trivial bundle of dimension , this generalizes the suspension isomorphism. In fact, a typical proof of this theorem for compact proceeds by induction over the number of open sets in a trivialization of , using the suspension isomorphism as the base case and the Mayer-Vietoris sequence to carry out the inductive step.
|Title||Thom isomorphism theorem|
|Date of creation||2013-03-22 15:40:52|
|Last modified on||2013-03-22 15:40:52|
|Last modified by||antonio (1116)|