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.
$H^1(X)$ 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.