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

5$\begingroup$ Look up the (incredible) KanThurston theorem. $\endgroup$ – Daniel Pomerleano Aug 19 '15 at 0:45

3$\begingroup$ Oh, I see Andy Putman already wrote that a minute earlier. $\endgroup$ – Daniel Pomerleano Aug 19 '15 at 0:47
The KanThurston theorem says that every pathconnected space is homology equivalent to an EilenbergMacLane 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.