TITLE: Classification of closed 3-manifolds with finite first homology group?
QUESTION [6 upvotes]: I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. 
Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, each closed 3-manifold with finite homotopy group has finite homology group. 
It is known that each closed 3-manifold with finite homotopy group $\Gamma$ is a spherical 3-manifold (i.e., is the orbit space $S^3/_\sim$ of the 3-sphere, endowed with a free action of the group $\Gamma$). 

Question. Is each closed 3-manifold with trivial homology group a spherical 3-manifold? Equivlalently, is the fundamental group $\pi_1(M)$ of a closed 3-manifold finite if its first homology group $H_1(M)$ is finite?

REPLY [6 votes]: Pick any knot in the three-sphere, and perform any Dehn surgery on it with some coefficient $p/q \neq 0$. This means that you remove the tubular neighborhood of the knot and you glue it back in a different way, parametrized by $p/q$. The manifold you get has $H_1(M,\mathbb Z) = \mathbb Z/_{p\mathbb Z}$. You get plenty of distinct 3-manifolds in this way. For instance, if the knot is hyperbolic, you get plenty of closed hyperbolic manifolds if $p$ or $q$ is sufficiently large. You can also require that $p=1$ and find plenty of homology spheres.