How to show there is no homotopy between two curves?

Question

On $\mathbb{C}\setminus\{0\}$, it seems that there can't be a homotopy between the curve given by $e^{i\theta},0\leq\theta<2\pi $ and the curve given by $e^{i\theta}+10,0\leq\theta<2\pi $. But if this is true, how does one prove this rigorously?

Answer 1

In $\Bbb C\setminus\{0\}$ they're not homotopic - the second curve is homotopic to a point and the first curve is not.

For example, $H\colon [0,2\pi[\times [0,1] \to \Bbb C \setminus \{0\}$ given by $H(\theta,s) = se^{i\theta}+10$ is such that $H(\theta,0) = 10$ for all $\theta$ and $H(\theta,1) = e^{i\theta}+10$ for all $\theta$.

The first curve is not homotopic to a point because, say, the line integral over it of the function $f(z) = 1/z$ is $2\pi i$ and not zero.