I've been reading John Lee's Introduction to Topological Manifolds and was having some trouble on the degree of the map $p_n:S^1\to S^1$ taking $z\mapsto z^n$. I understand his proof of why the degree of $p_n$ is $n$, but I've somehow managed to come up with an alternate (and obviously invalid) "proof" for why the degree is also $0$:
We use Problem 8-4, which states that a continuous map $\phi:S^1\to S^1$ has an extension to a continuous map $\Phi:\bar B^2\to S^1$ if and only if it has degree zero. Since $p_n$ can be extended to $\overline B^2$ by the obvious map taking $z\in\overline B^2$ to $z^n$, it follows that the degree of $p_n$ must be $0$.
This feels like a pretty stupid question to ask, so I'm pretty sure I'm just missing something really obvious. If someone could explain where the issue in my "proof" lies, I'd really appreciate it!
Your extended $p_n$ does not have values in the circle anymore. The codomain is wrong.