Let $M$ be a connected topological manifold with universal covering $\pi: \widetilde{M} \rightarrow M$ and let $p \in \widetilde{M}$ be a point.
Let $\alpha,\beta : \widetilde{M} \rightarrow \widetilde{M}$ be two deck transformations.
Let $\gamma_1 \subset \widetilde{M}$ be a path connecting $p$ and $\alpha \circ \beta(p)$ and let $\gamma_2 \subset \widetilde{M}$ be a path connecting $p$ and $\beta \circ \alpha(p)$.
My question is: is the homology class of the loop $\pi \circ \gamma_1$ the same as the homology class of the loop $\pi \circ \gamma_2$ in $M$. I have the feeling that this should be false for the classes in the fundamental group and is connected the the homology group being abelian.
Thanks :)