For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$$
The above map is as in https://mathoverflow.net/questions/202976/interpretation-of-the-monomorphism-h2-pi-1x-mathbbz-rightarrow-h2x/207001?noredirect=1#comment513195_207001.