I am reading a homotopical proof of the fundamental theorem of algebra, and the end of the proof is as follows:
... the map $z\mapsto r^nz^n$ (where $r$ is a positive real number and $z\in S^1$) is homotopic to a constant map $z\mapsto a_0$ as maps from $S^1$ to $\mathbb R^2-0$. But $\mathbb R^2-0\cong S^1$,...
So does it follow from here that the maps from $S^1$ to itself given by $z\mapsto z^n$ and $z\mapsto c$ for a constant $c$ are homotopic, and since the fundamental group of $S^1$ is $\mathbb Z$, the first map induces a nonzero map while the constant map induces the zero map which is contradictory?