group cohomology (topological definition)
Let be a topological group. Suppose some contractible space admits a fixed point free action of , so that the quotient map is a fibre map. Then , denoted is called the classifying space of . Classifying spaces always exist and are unique up to homotopy. Further, if has the structure of a CW- complex, we can choose to have one too.
The group (co)homology of is defined to be the (co)homology of . From the long-exact sequence associated to the fibre map, , we know that for . In particular the fundamental group of is , which inherits a group structure as a quotient of . Let denote . Then acts freely on the cells of , the universal over of . Hence the cellular resolution for , denoted, , is a sequence of free - modules and - linear maps. Taking coefficients in some - module , we have
In particular, when is discrete, must be the covering map associated to a universal cover. Hence and is exact, as is contractible and hence has trivial homology. Note in this case . So for a discrete group , we have,
As an example take . Note topologically, . As for , we know that .
More explicitly, we may identify with the unit complex numbers. This acts freely on the infinite complex sphere (which is contractible) leaving a quotient of .
Hence if divides and otherwise.
Similiarly and , as and are isomorphic to U(R) and U(H) respectively. So for all and if divides and otherwise.
|Title||group cohomology (topological definition)|
|Date of creation||2013-03-22 14:32:24|
|Last modified on||2013-03-22 14:32:24|
|Last modified by||whm22 (2009)|