Is anyone aware of a comprehensible example of a topological space $X$ with $\pi_1(X) \neq 0$ and $H_1(X)=0$? We could always build a space with a perfect fundamental group, such as $A_5$, and so arrive at an example, but perhaps it would be very difficult to understand. I'm hoping someone could help me understand how a space can have nonzero fundamental group but zero homology group.

I guess the Poincare Homology Sphere has fundamental group $A_5$ (although some sources seem to claim the group is of order 120...maybe there is differing terminology going on here). Maybe that's a start.

  • $\begingroup$ My sources tell me that the fundamental group of the Poincaré homology sphere is the binary icosahedral group of order $120$, but note that this is also a perfect group. $\endgroup$ – Slade Oct 25 '15 at 20:16
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    $\begingroup$ It's not a start, it's an end: the Poincare homology sphere has fundamental group $\widetilde{A}_5$ (not $A_5$ but a particular central extension of it), which is perfect. You're already done now. $\endgroup$ – Qiaochu Yuan Oct 25 '15 at 20:50

$H^1(X)$ is $\pi_1(X)/[\pi_1(X),\pi_1(X)]$, you can take a non abelian simple discrete group $G$ and $X$, $K(G,1)$ the $1$-Eilenberg-McLane space.

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  • $\begingroup$ $G$ must be non-abelian, right? $\endgroup$ – M.U. Oct 25 '15 at 20:11

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