Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf
I don't understand the proof of Lemma 5.1.
Notation: $T_{0,0}$ is the 2-sphere, $T_{g,0}$ is a surface of genus 0 with 0 punctures, $T_{g,n}$ is a surface of genus g with n punctures.
Lemma 5.1 Let $(p, T_{g,0}, T_{0,0})$ be a cyclic branched covering. Let $(\tilde{p}, T_{g,n}, T_{0,n})$ be the associated unbranched covering. Then every homeomorphism of $T_{0,n}$ lifts to a homeomorphism of $T_{g,n}$. (only unique up to covering transformation)
The proof is short yet I am having trouble filling in the details for:
Why since the covering is k-sheeted and cyclic, a closed curve lifts to a closed curve if and only if it encircles a multiple of k branch points. In particular how is the fact that it is a cyclic covering used.
Any help or suggestions will be warmly received. thanks
Suppose your curve $C$ encircles $b$ branch points (counted with multiplicity, if it encircles the same branch point more than once or in opposite directions). Replace it with a homotopic curve $C'$ that returns to the base point after each encircling of a branch point. That is, $C'$ is a concatenation of curves $D_i$ that each encircle a branch point exactly once, in the positive or negative direction. (The number of $D_i$ that encircle branch points positively minus the number that encircle branch points negatively is $b$.) Since $C$ and $C'$ are homotopic with fixed endpoints, one of them lifts to a closed curve if and only if the other does. So it suffices to consider $C'$. Lifting $C'$ amounts to lifting each of the $D_i$'s in turn, starting each one where the previous one ended. Each $D_i$ that encircles a branch point positively (resp. negatively) has a lifting that ends one sheet "higher" (resp. "lower") than it began, where "higher" and "lower" refer to adding or subtracting $1$ in the cyclic structure. (The details of this depend on exactly how your source defines "cyclic covering" --- I hope it wasn't defined as "curves lift to closed curves iff they encircle branch points a net multiple of $k$ times" because then all this was a waste of time.) So for the lifts of all the $D_i$ together to return to the original starting point, we need that the number of additions of $1$ minus the number of subtractions of $1$ (the net upward or downward motion of the covering curve) is a multiple of $k$.