Winding number and lifts I am working with the following paper https://arxiv.org/abs/math/0405583.
In chapter 7, proof of Lemma 7.2 it states the following.
Let $D^2$ be the 2 dimensional unit disk and fix a finite set $S\subset \overset{\circ}{D}$. Let $X$ be the ramified double cover of $D$ with ramification points given by $S$. Let $\gamma$ be a loop in $D$ then the following holds.
The number of points with odd winding number for $\gamma$ is even $\iff$ the lift of $\gamma$ to $X$ closes up.
I do not see how this can be proven and and didn't found any references for this.
I am very thankful for any kind of hint.
 A: Thanks for your hint @Warlock of Firetop Mountain. I have made up an idea in a more general setting.
Suppose we are given a surface $S$ and a ramified double cover of $S$ with a finite number of ramification points, say $R$. Fix one of these points $b\in R$. Let $\gamma$ be a loop in $S\setminus R$ that encircles $b$, but no other points of $R$. 
Recall that we can think of a 2-sheeted ramified cover, as a 2-covering where the two sheets are identified at the ramification point. Label the two sheets by $C_1$ and $C_2$. Fix a basepoint $p$ of $\gamma$ and let $\overline{p}$ be its lift in the cover (since the loop contains no points of $R$, the branched covering reduces to an usual covering). Without loss of generality, $\overline{p}\in C_1$. Now every time the loop $\gamma$ winds around $b$, the lift of alpha passes from one sheet of the covering to the other. Thus if $b$ has odd winding number with respect to the loop, the lift of $\gamma$ will end in $C_2$ and is thus not closed. Conversely if $\gamma$ winds an even number of times around $b$, it ends in $C_1$ and closes up.
In my opinion, provided my result is true, the original question should now directly follow from this by splitting the loop in parts, such that each part encircles only one ramification point. 
Do you think this works, or are there any hints to improve this?
