One more problem preparing for a PhD exam! It states "describe a space such that $H^1(X,Z)=Z_5$." I thought this was impossible by the universal coefficient theorem since $H_0(X;Z)$ is always free, and $HOM(H_1(X,Z),Z)$ is free then $H^1(X,Z)$ is also free. Any help would be great. Thanks!
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You're correct. Perhaps the intended problem was about $H_1$. – Qiaochu Yuan Apr 29 '16 at 00:59
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Thank you! That makes sense. That problem is a lot easier :) – chachoman Apr 29 '16 at 01:48