TITLE: Group cohomology intuition
QUESTION [17 upvotes]: Group cohomology is one thing I don't seem to get my head around.
I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.
For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!
Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places?

REPLY [8 votes]: Group cohomology can be defined very naturally in a purely topological way. The definition of $1$-cocycles is not random, or due to historical accident.
More specifically, given a group, $G$, the Eilenberg-Maclane space $X=K(G,1)$ is defined which has $\pi_1(X)=G$, and $\pi_{\geq 2}(X)=0$. This is well-defined up to homotopy-type if you assume it is a CW-complex. Since the cohomology functor is invariant under homotopy equivalence, the groups $H^i(X)$ are well-defined abelian groups associated with the original group $G$, and this is what we call group cohomology. Now when you take coefficients in a $\mathbb Z[G]$ module $M$, this is actually equivalent to taking cohomology with ``twisted coefficients" of $K(G,1)$.