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Sep 5, 2017 at 2:21 comment added Nicholas Kuhn I'm most familiar with the mod p cohomology of $K(\mathbb Z/p,n)$'s, where the image of $e^*$ does generate the cohomology ring. But I am dubious that this holds integrally.
Sep 5, 2017 at 2:07 comment added Vitali Kapovitch I was hoping my question can be answered without explicitly computing $H^*(K(\mathbb Z), n),\mathbb Z)$ which I know is hard to compute. Perhaps by induction on $n$ by analyzing the spectral sequence of $K(\mathbb Z,n)\to\star\to K(\mathbb Z,n+1)$.
Sep 5, 2017 at 1:31 history answered Nicholas Kuhn CC BY-SA 3.0