Cohomology of Topological Groups and Grothendieck Topologies

Igor Minevich

December 4, 2014

For an abstract group G, there is only one "canonical" theory H^n(G, A) of group cohomology for a G-module A. There are many different ways to generalize this theory when G is a topological group and A is a topological G-module. Some examples are the continuous cochain theory, Moore's measurable cochain theory, and Wigner's semisimplicial theory. Lichtenbaum was able to use the semisimplicial theory to state a conjecture about special values of the zeta function of a number field. The questions then arise: (1) how do we compare the different theories, and (2) can we apply other theories just as Lichtenbaum did? We will talk about how Grothendieck topologies can help answer these questions.