Homology of universal abelian cover of a manifold

Question

If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, then what can one say about $H_1(M_0)$ ? Note that Hurewicz Theorem gives us a group isomorphism between $H_1(M_0)$ and the abelianization of $[\pi_1(M),\pi_1(M)]$.

In particular, I would like to understand why is the following integral independent of the choice of a $C^1$ curve $\tau$ in $M_0$ with fixed endpoints: $$ \int_\tau \overline{\omega}, $$ where $\overline{\omega}$ is the lift of a closed 1-form $\omega$ on $M$.

Answer 1

This has not much to do with $H_1(M_0)$. If $\pi :M_0\rightarrow M$ is your abelian covering, we have $\int_{\tau }\overline{\omega} =\int_{\pi _*\tau }\omega $. But the exact sequence $0\rightarrow \pi _1(M_{0})\rightarrow \pi _1(M)\rightarrow H_1(M)\rightarrow 0$ shows that $\pi _*\tau $ is zero in $H_1(M)$, hence the integral is zero.