It many sources it's stated that the winding number is invariant under homotopy, but I've yet to actually see why.
Suppose you have the formal definition of the winding number. So for a continuous loop $\gamma\colon[\alpha,\beta]\to\mathbb{C}\setminus\{a\}$ which doesn't pass through a point $a$, one has the function $\theta(t)=\text{arg}(\gamma(t)-a)\in\mathbb{R}/2\pi\mathbb{Z}$. By the lifting lemma, there exists a continuous $\tilde{\theta}\colon[\alpha,\beta]\to\mathbb{R}$, such that $[\tilde{\theta}(t)]=\theta(t)$, and the winding number of $\gamma$ around $a$ is then defined as $$n(\gamma,a)=\frac{\tilde{\theta}(\beta)-\tilde{\theta}(\alpha)}{2\pi}.$$
Is there a straightforward proof that the winding number is invariant under homotopy with this definition for continuous loops which do not pass through $a$? Thanks.
Best Answer
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.