Why is the winding number homotopy invariant?
If you have a homotopy $H : [\alpha, \beta] \times [0; 1] \to \mathbb{C}\setminus \{a \}$, the function $\theta : [\alpha, \beta] \times [0; 1] \to \mathbb{R}/2 \pi \mathbb{Z}$ defined by $\theta(x,t) = \arg(H(x,t)-a)$ is continuous in both variables, and can be lifted into a function $\tilde{\theta}$.
Then you can define the continuous function $n(t) = \frac {\tilde \theta (\beta,t) - \tilde \theta (\alpha,t)}{2 \pi} \in \mathbb{Z}$. Since $\mathbb{Z}$ is discrete, it has to be a constant map, so the winding numbers of $\gamma = H(.,0)$ and $H(.,1)$ are the same.