# Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy answer, or spectral sequences over $\mathbb{Z}$ are not as simple as over $\mathbb{Z}_2$?

• See doc.rero.ch/record/482/files/Clement_these.pdf (including tables in Appendix C). In particular, yes, $H^5(K(Z/2,2))$ has 4-torsion, and no, I don't think there is a really simple and explicit answer. – Grigory M Aug 10 '15 at 12:32
• @Grigory, thank you, i'll do it in a few days – Andrey Ryabichev Aug 10 '15 at 13:07