TITLE: Examples of non-simply connected manifolds with trivial H^1 
QUESTION [12 upvotes]: It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can construct a CW-complex, via generators and relations, having G as a fundamental group. Are there such examples in the class of topological or differentiable manifolds? In other words, does there exist a non-simply connected manifolds with trivial first homology group?

REPLY [8 votes]: Fake Projective planes
These are smooth complex projective surfaces with the same betti numbers as $\mathbb{CP}^2$, but with infinite fundamental group $\pi_1(X)$ (in fact it is isomorphic to a torsion-free cocompact arithmetic subgroup of $PU(2,1)$).