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!
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5$\begingroup$ Look up the (incredible) Kan-Thurston theorem. $\endgroup$– Daniel PomerleanoAug 19, 2015 at 0:45
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3$\begingroup$ Oh, I see Andy Putman already wrote that a minute earlier. $\endgroup$– Daniel PomerleanoAug 19, 2015 at 0:47
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1 Answer
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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.
answered Aug 19, 2015 at 0:44