Group Cohomology - Connections With Topological Cohomology Theories

Connections With Topological Cohomology Theories

Group cohomology can be related to topological cohomology theories: to the topological group G there is an associated classifying space BG. (If G has no topology about which we care, then we assign the discrete topology to G. In this case, BG is an Eilenberg-MacLane space K(G,1), whose fundamental group is G and whose higher homotopy groups vanish). The n-th cohomology of BG, with coefficients in M (in the topological sense), is the same as the group cohomology of G with coefficients in M. This will involve a local coefficient system unless M is a trivial G-module. The connection holds because the total space EG is contractible, so its chain complex forms a projective resolution of M. These connections are explained in (Adem-Milgram 2004), Chapter II.

When M is a ring with trivial G-action, we inherit good properties which are familiar from the topological context: in particular, there is a cup product under which

is a graded module, and a Künneth formula applies.

If, furthermore, M = k is a field, then is a graded k-algebra. In this case, the Künneth formula yields

For example, let G be the group with two elements, under the discrete topology. The real projective space is a classifying space for G. Let k = F₂, the field of two elements. Then

a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of .

Hence, as a second example, if G is an elementary abelian 2-group of rank r, and k = F₂, then the Künneth formula gives

,

a polynomial k-algebra generated by r classes in .