$f: S^n \to S^n$ is the restriction of a continuous mapping $F: \overline{B^n_1} \to S^n$ iff $deg(f) = 0$.

Question

this is a theorem from wikipedia (https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping#Properties):

A self-map $f: S^n \to S^n$ of the $n$-sphere is extendable to a (continuous) map $F: B_n \to S^n$ from the $n$-ball to the $n$-sphere if and only if $\text{deg}(f) = 0$. (Here the function $F$ extends $f$ in the sense that $f$ ist the restriction of $F$ to $S^n$)

I wasn't able to show it and couldn't find a proof. I'd appreciate any help!

Answer 1

If there is such an extension, then $H_n(f)$ factors over $H_n(D^n)=0$. If $\deg(f)=0$, then $f$ is zero in $\pi_n(S^n)$ by Hurewicz, hence extends to a map on the disk.