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What is the cohomology/homology of $K(\mathbb{Z},n)$? In my algebraic topology class today, we computed the cohomology of $K(\mathbb{Z},2)$ and showed that the space is equivalent to $\textbf{CP}^\infty$. Can we make similar statements about $K(\mathbb{Z},n)$ for higher $n$? I tried to compute the cohomology of $K(\mathbb{Z},3)$ and managed to show that $H^3(K(\mathbb{Z},3))=\mathbb{Z}$ and $H^6(K(\mathbb{Z},3))=\mathbb{Z}/2$, but got stuck after that. Is anything else known about these spaces? What are some papers that I could look at?

Dario
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ruadath
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  • The keyword you want is "cohomology operations." Computing these is equivalent to computing the cohomology of Eilenberg-MacLane spaces. – Qiaochu Yuan Feb 21 '15 at 06:27
  • I'm not sure what you mean. Could you elaborate? – ruadath Feb 21 '15 at 06:30
  • Cohomology operations are natural transformations between cohomology functors, and by the Yoneda lemma these are equivalent to classes in the cohomology of Eilenberg-MacLane spaces. If you want to look up computations of the cohomology of Eilenberg-MacLane spaces, one way to do it is to look up computations of (unstable) cohomology operations, e.g. in Mosher-Tangora. – Qiaochu Yuan Feb 21 '15 at 06:32
  • I think things are pretty complicated over $\mathbb{Z}$ though. For example $H^8(B^3 \mathbb{Z}, \mathbb{Z})$ turns out to be $\mathbb{Z}_3$; I asked a question about it here: http://math.stackexchange.com/questions/422493/the-simplest-nontrivial-unstable-integral-cohomology-operation. – Qiaochu Yuan Feb 21 '15 at 06:34

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