Eilenberg MacLane space with nonzero homology

Question

Does there exists an abelian group $G$ such that $H_2(K(G,1);\mathbb{Z})\neq 0$? If so, what is an example of such a group $G$?

The question is motivated by this post.

Answer 1

The torus $T^2$ is a $K(\Bbb Z^2,1)$ with nonzero second homology, since it is an orientable 2-manifold.

Answer 2

The singular homology of the Eilenberg-Mac Lane space $K(G,1)$ is isomorphic to group homology $$H_n(K(G,1);\Bbb Z)\cong H_n(G;\Bbb Z).$$ Joyner's paper on computational group (co)homology theory gives as an example $$H_2(A_5;\Bbb Z)\cong\Bbb Z/2\Bbb Z.$$

Answer 3

For $G = \mathbb{Z}_n \oplus \mathbb{Z}_n$, the group $H_2(K(G, 1)) = H_2(G)$ contains a term $H_1(\mathbb{Z}_n)^{\otimes 2}\not= 0$ by the Kunneth formula. If you don't require that $G$ is finite, just take $G = \mathbb{Z}^n$ and $K(G, 1) = T^n$ for $n\geq 2$.