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 $M$.



Homology of universal abelian cover of a manifold
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.


