If one define the universal abelian cover $M_0$ of a manifold $M$ as the abelian cover (i.e. normal cover with abelian group of deck transformations) that covers any other abelian cover, 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 the $C^1$ curve $\tau$ in $M_0$: $$ \int_\tau \overline{\omega}, $$ where $\overline{\omega}$ is the lift of a closed 1-form $\omega$ on $M$.