# Homology of universal abelian cover of a manifold

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 $$M$$.

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.