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?
Best Answer
The answer is no by Yves' comments. Let me add that there are plenty of explicit constructions of closed hyperbolic 3--manifolds with finite homology, and this is a generic phenomenon (for example random Heegard gluings have zero first Betti number and are hyperbolic and numerical experiments on the census manifolds exhibit an overwhelming proportion of manifolds with zero first Betti number). This hints to there being no hope to get a classification.
For various results about hyperbolic rational homology spheres (probabilistic, numerical, explicit constructions of infinite families) see for example the papers of Nathan Dunfield and coauthors: