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 $M$.