Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$
again I can only do one direction
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$f:S^1→S^1$ is smooth, and $S^1 = \partial B$. Assume that $f:S^1→S^1$ extends to the whole ball$B$ then $deg(f)=0$ by the boundary theorem.
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Assume that $deg(f)=0$. The book tell me to extend $f$ to the annulus $A=\{\frac 12 \leq |z| \leq 1\}$ so that the inner circle $\{|z|= \frac 12\}$, the extended map is constant. And again I have no idea how this can be helpful to me.
If $\deg(f) = 0$, then $f$ is null-homotopic; choose a homotopy $F_t:I \times S^1 \to S^1$ between $F_0 \equiv 1$ and $F_1 = f$. Define $\tilde f$ on $A$ by $\tilde f(r, \theta) = f_{2r - 1}(\theta)$ in polar coordinates. Since $\tilde f$ is constant on the circle $\{z:\, |z| = \frac{1}{2}\}$, it's trivial to extend it over the rest of the disk.