Sebastian Thomas (Aachen), On the second cohomology group of a simplicial group.

Abstract:

To a group G, we can assign a path-connected topological space BG whose (singular) cohomology coincides with the group cohomology of G. The fundamental group of BG is pi_1(BG) = G and the higher homotopy groups pi_n(BG) for n >= 2 are trivial. Conversely, every path-connected topological space X with pi_n(X) = {1} for all n >= 2 is weakly homotopy equivalent to B pi_1(X). In this sense, groups are algebraic models for path-connected topological spaces whose pi_n vanishes for n >= 2.

We generalise this situation: A crossed module V consists of a group Gp V, a group Mp V, a group homomorphism mu^V: Mp V -> Gp V and a compatible group action of Gp V on Mp V. An example of a crossed module is given by the inclusion N -> G of a normal subgroup N in a group G. Another example is given by the projection E -> G of a central extension E of a group G. Crossed modules are algebraic models for path-connected topological spaces whose pi_n vanishes for n >= 3. A group G can be considered as the particular crossed module {1} -> G.

Still more generally, simplicial groups model arbitrary path-connected topological spaces, with no vanishing condition on the homotopy groups pi_n. Crossed modules can be considered as particular simplicial groups.

Eilenberg and Mac Lane gave a pullback description for the second cohomology group H^2(X) of an arcwise connected topological space X, which involves the homotopy groups pi_1(X) and pi_2(X) and an element k^3_X of the group cohomology group H^3(pi_1(X), pi_2(X)). We show how the simplicial group version of Eilenberg's and Mac Lane's theorem can be proven by purely algebraic methods, using a reduction to the case of crossed modules.