Covering space of a $3$-manifold with infinite fundamental group

Question

For a $3$-manifold $X$, s.t. $\pi_1(X)$ is infinite, how to see the universal covering space of $X$, $\tilde{X}$ is a non-compact $3$-manifold and $H_3(\tilde{X})=0$?

Answer 1

I assume $X$ is intended to be connected. If $x\in X$ and $p:\tilde{X}\to X$ is the covering map, then $p^{-1}(\{x\})$ is closed in $\tilde{X}$, discrete (since $p$ is a covering map), and in bijection with $\pi_1(X)$. This means $\tilde{X}$ is not compact, since it has a closed subset which is not compact. The fact that $H_3(\tilde{X})=0$ then follows from the fact that $\tilde{X}$ is a noncompact connected $3$-manifold (see for instance Proposition 3.29 of Hatcher's Algebraic Topology).