I am looking for a good reference for the structure of the cohomology ring $H^*(Z^n,Z)$. In particular, I would like to know how large is the subgroup of $H^2(Z^n,Z)$ generated by cup-products from $H^1(Z^n,Z)$.
I will be grateful for any help!
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3The group cohomology of $\mathbb{Z}^n$ with trivial coefficients agrees with the cohomology of its classifying space, which is the $n$-torus $T^n$. So the cohomology is the exterior algebra on $H^1$. – Qiaochu Yuan Apr 09 '15 at 18:32
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The Kunneth formula holds here by abstract nonsense, and the cross product maps are isomorphisms because $H^1(\mathbb{Z}, \mathbb{Z}) = H^1(S^1, \mathbb{Z})$ is torsion-free. – anomaly Apr 09 '15 at 19:16