Let $\gamma_0(t) = e^{2\pi it}$ and $\gamma_1(t) = e^{-2\pi it}$ for $0\le t\le 1$.
I have to check whether or not they are homotopic in $\Bbb C -\{0\}$.
I can see that the index of $n(\gamma_0;0)=1$ and $n(\gamma_1;0)=-1$, how can i use this to check their homotopy ?
Also i know the mathematical definiton of homotopy but how can i check it geometrically in this case ,both $\gamma_0(t)$ and $\gamma_1(t)$ will be in opposite directions on the same circle, so can i conclude from this that they won't be homotopic to zero ?
Assume there is a homotopy $\gamma_t : [0,1]\to\mathbb C\setminus\{0\}$, $t\in [0,1]$, then $$ f(t)=\mathrm{Ind}_{\gamma_t}(0)=\frac{1}{2\pi i}\int_{\gamma_t}\frac{dz}{z}, $$ would be a continuous function of $t$ which takes only integer values. Hence $f$ is constant, and thus $$\mathrm{Ind}_{\gamma_0}(0)=\mathrm{Ind}_{\gamma_1}(0). $$ Contradiction.