Q: homotopy in \mathbb{C}-{0} Is it true that the identity map from $\mathbb{C}-\{0\} \to \mathbb{C}-\{0\}$ is homotopic to any continuous map $f:\mathbb{C}-\{0\} \to S^{1}$  where $S^{1}$ is unit circle in $\mathbb{C}$ ?
PS: I showed if $f(z)=\frac{z}{||z||}$ then both are homotopic, so eager to know whether its true for any general such $f$ or not. Someone told me yes.

A: No, for example take the map $z\mapsto z^2$ on $\mathbb{C} \setminus \{0\}$ and compose with the projection $z \mapsto z/\lVert z\rVert$. If you're already familiar with the fundamental group of $S^1$, then the it should be clear that these are not homotopic. Or you could use winding numbers after composing with the inclusion $S^1 \to \mathbb{C}$.
