Recently I came across the following question: can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the integral cohomology ring of some Eilenberg-Maclane space $K(\pi,1)$? I guess (without strong evidences) that the answer is negative. If so, how to prove it? Thanks in advance!

## 1 Answer

The Kan-Thurston theorem says that every path-connected space is homology equivalent to an Eilenberg-MacLane space, so the answer is "yes". See

Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976.