Regard $\mathbb Z^n$ as an abelian group, and let $\mathbb C^* = \mathbb C-\{0\}$.
Question: What is the group cohomology $H^2(\mathbb Z^n, \mathbb C^*)$?
More specific question is as follows. I have found out that for $n=2$, a map $f: \mathbb Z^2 \times \mathbb Z^2 \to \mathbb C^*$, defined as $$f((m_1, m_2), (n_1, n_2)) = (-1)^{m_2n_1},$$ becomes a 2-cocyle, by explicitly checking all cases in Mathematica. Is $f$ 2-coboundary? (The motivation for finding an example $f$ comes from the "Jordan-Wigner string" in physics.)
Edit: The 2nd group cohomology $H^2(G,A)$ is related to the central extension of $G$ by an abelian group $A$. Define $$Z_2(G,A) = \{ \omega: G\times G \to A: \omega(x,y) \omega(xy, z) = \omega(x, yz) \omega(y,z), \omega(1,1)=1\}.$$ Also, for a function $\lambda: G \to A$ with $\lambda(e)=e$, define $\omega_\lambda(x,y) = \lambda(x) \lambda(y) \lambda(xy)^{-1}$. Also, define $$B_2(G,A) = \{\omega_\lambda: G \times G \to A\}_{\lambda:G\to A}.$$ The second cohomology is defined as $H^2(G,A) = Z_2(G,A)/B_2(G,A)$.