Lifting homeomorphisms covering 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
 A: This response is long overdue, but I just saw this message a few months ago and I wanted to check with Birman and Hilden before responding.
The lemma is actually incorrect exactly because of the line you pointed out.  However, there is a certain family of cyclic branched covers of the sphere where it is true, which I describe below.  Ty Ghaswala and I noticed this error while writing http://arxiv.org/abs/1604.03908, we contacted Birman and Hilden and they wrote an erratum that has been accepted to the Annals.  It should be posted on Joan Birman's webpage in the next few weeks (the ArXiv cannot accept errata).  Then Ghaswala and I wrote a follow up paper: http://arxiv.org/abs/1607.06060 that exactly characterizes when the lemma is true, which is basically what I will describe below.
Let $S_0$ be the sphere (it has branch points) and $k$ the degree of the cover.  When the loops around each branch point maps to the same number under $H_1(S_0,\mathbb{Z})\rightarrow \mathbb{Z}/k\mathbb{Z}$, then the curves that bound $k$ branch points are exactly the curves that lift and the proof works as written.  
I think for a long time there was confusion, at least within topology, about the uniqueness of cyclic branched covers of the sphere.  I am still at little confused from this perspective because the non-uniqueness of cyclic branched covers seems to contradict a result of L\"{u}roth.  However, from the algebraic geometry perspective, a cyclic branched cover of the sphere is a solution to an equation of the form $$y^k=(x-a_1)^{p_1}\cdots(x-a_n)^{p_n}.$$  Here the $a_i$ are the branch points and $p_i$ is the image of a loop isotopic to $a_i$ under the map $H_1(S_0,\mathbb{Z})\rightarrow \mathbb{Z}/k\mathbb{Z}$.  While the coefficients do not unique determine the cover, they do distinguish covers in a way that I think is much more clear than in topology (see "Groups as Galois Groups" by V\"{o}lkein as a reference).
I hope this helps, please let me know if I need to clarify anything.  I tried to summarize briefly both the correct solution and the root of confusion, but I hope it was not too brief.
A: 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 $k$.
