[Math] Continuous function on unit circle has fixed point

analysisfixed-point-theoremsfunctions

The question I have is:

Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.

Best Answer

$f$ is not ONTO,this is nul homotopic, so it will have fixed point.

let $f:S^1\rightarrow S^1$ be nulhomotopic, then it extends to a map $F$ from $B^2$ to $S^1$ which can be thought of as a map from $B^2$ to $B^2$, and then apply Brouwer's Fixed Point Theorem, $F$ must have a fixed point,. Since the image of $F$ is contained in $S^1$ so this fixed point must lie in $S^1$ so it is also a fixed point of $f$.

Also we see as this is null homotopic so degree of $f=0$, but there is a result saying $f:S^1\rightarrow S^1$ with no fixed point has degree $(-1)^{n+1}$

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