Homotopy

Let $X = \mathbb{C}\setminus\{1, -1\}$ with basepoint $0$. For a loop $\gamma$ in $X$ let $w_{\pm}(\gamma)$ be the degree of $(\gamma - (\pm1))/|\gamma - (\pm1)|$, the winding number of $\gamma$ about $\pm1$. (a) Show that $W : \pi_1(X, 0)\to\mathbb{Z}^2$, $[\gamma]\mapsto(w_+(\gamma), w_-(\gamma))$, is a well-defined homomorphism. (b) Show that $W$ is surjective. (c) Deduce that $X$ is not contractible and not homotopy equivalent to $S^1$.