It is well known that the singular (co)homology of the space $K(G,1)$ is one possible definition of group homology. Now for non-abelian groups, we cannot expect to extract anymore obvious group invariants using algebraic topology.
But for abelian group, we get a bigraded collection of invariants, by considering the homology of higher Eilenberg MacLane spaces. My question is whether this leads to any interesting.
I know this is a very soft question, and so I understand that the answers might be equally soft.