$\newcommand\C{\mathbb C} \newcommand\bs{\backslash}$There is a book on Riemann surfaces by Forster that I like very much. I am curious about the first exercise in chapter 1 section 4 on branched and unbranched coverings:
Let $X:=\C\bs\{\pm1\}$ and $Y:=\C\bs\{(2k+1)\pi/2 : k\in \mathbb Z\}$. Show that $\sin: Y\rightarrow X$ is a topological covering map. Consider the following curves
$$u:[0,1]\rightarrow X \qquad t\mapsto 1-e^{2\pi i t}$$ $$v:[0,1]\rightarrow X \qquad t\mapsto -1+e^{2\pi i t}$$and let $w_1:[0,1]\rightarrow Y$ be the lifting of $u\cdot v$ with $w_1(0)=0$ and $w_2:[0,1]\rightarrow Y$ be the lifting of $v\cdot u$ with $w_2(0)=0$. Show that ($w_1$ and $w_2$ are not "loops") $w_1(1)=2\pi$ ad $w_2(1)=-2\pi$ (and with this one concludes that the fundamental group $\pi_1(X)$ is non-abelian).
I think I am able to do all this, but I think what I get is that $w_1(1)=4\pi$ and $w_2(1)=-4\pi$ (I tried to visualize the whole thing graphically, which is not easy). Is it really the case that $w_1(1)=2\pi$ and $w_2(1)=-2\pi$ or did I just found an error in the book?