Let $M$ be prime closed compact orientable 3-manifold with infinite fundamental group $G=\pi_1(M)$. Can there be finite order elements in $G$ ? What are the possibilities for $H_1(M)$ - can it be any abelian finitely generated group ?
If we have finite cover $N\to M$. In this case there is monomorphism $\pi_1(N)\to\pi_1(M)$. Is there also monomorphism $H_1(N)\to H_1(M)$ ?
From the "virtually Haken conjecture" I conclude that there exists finite cover $N$ which is Haken. It means it contains incompressible surface. It means $H_2(N)\neq0$.
If $M$ is closed compact orientable 3-manifold with finite fundamental group then $M$ is spherical. It means it is quotient of $S_3$ by finite subgroup of $SO_4$. This subgroup is fundamental group of $M$.