On page 23 of "A primer on Mapping Class Groups" by Farb and Margalit, they state the following:
Now suppose that $\widetilde{S}$ is the universal cover (of the closed, connected, orientable surface $S$) and $\alpha$ is a simple closed curve in $S$ that is not a nontrivial multiple of another closed curve. In this case, the lifts of $\alpha$ to $\widetilde{S}$ are in natural bijection with the cosets in $\pi_1(S)$ of the infinite cyclic subgroup $\langle\alpha\rangle$. (Any nontrivial multiple of $\alpha$ has the same set of lifts as $\alpha$ but more cosets.) The group $\pi_1(S)$ acts on the set of lifts of $\alpha$ by deck transformations, and this action agrees with the usual left action of $\pi_1(S)$ on the cosets of $\langle\alpha\rangle$. The stabilizer of the lift corresponding to the coset $\gamma\langle\alpha\rangle$ is the cyclic group $\langle\gamma\alpha\gamma^{-1}\rangle$.
They define a lift of a curve $\alpha : S^1 \to S$ as a lift of the map $\alpha \circ \pi : \mathbb{R} \to S$ to $\widetilde{S}$, where $\pi: \mathbb{R} \to S^1$ is the usual covering.
I cannot understand how the set of lifts is in bijection with the set of cosets. Given a coset $\gamma\langle\alpha\rangle$, are we considering the lift of the loop $\gamma \circ \alpha$ as another lift of $\alpha$? And what should be the map the other way around? I can understand how the other remarks follow if I grant the existence of this bijection. Could someone tell me how the bijection is obtained?
Best Answer
Since $p: \widetilde{S} \to S$ is the universal cover, $\pi_1(S)$ acts on $\widetilde{S}$ by deck transformations: $\gamma \cdot \tilde{s}$ is the endpoint of the lift of $\gamma^{-1} \in \pi_1(S, p(\tilde s))$ starting at $\tilde s$ (the inverse is to make it a left action). Since these are deck transformations, ie $p \circ \gamma = p$, it follows that $\gamma(\widetilde \alpha) = \gamma \circ \widetilde \alpha$ is a lift of $\alpha$ whenever $\widetilde \alpha$ is. By unique lifts after choosing basepoints, once you fix one lift $\widetilde \alpha$, every lift is of this form as $\gamma$ varies in $\pi_1(S)$.
Now it suffices to check that this is compatible with the action of $\gamma$ on the coset $\langle \alpha \rangle \subset \pi_1(S)$.