# can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the cohomology of some Eilenberg-Maclane space $K(\pi,1)$?

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!

• Look up the (incredible) Kan-Thurston theorem. – Daniel Pomerleano Aug 19 '15 at 0:45
• Oh, I see Andy Putman already wrote that a minute earlier. – Daniel Pomerleano Aug 19 '15 at 0:47