every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point?

Question

can someone please help me to show for every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point? i think since the fuction is not surjective it doesn't contain at least one point of $S^2$ so it's like a function from $S^2$ to $R^2$ and by Borsuk-Ulam there exist a point x s.t. f(x)=f(-x) but i'm not sure if this can help...

Answer 1

Here are two ways:

You can use your intuition. Indeed, a nonsurjective map $f:S^2 \to S^2$ factors through a map $\tilde{f}:S^2 \to \mathbb R^2$, and hence it must be nullhomotopic.

In particular, $f_*:H_2(S^2,\mathbb R) \to H_2(S^2,\mathbb R)$ is the zero map.

Consequently, one can apply the Lefschetz-fixed point theorem to calculate the lefschetz number

$\Lambda_f(S^2)=\sum_{i=0}^{2}(-1)^i\mathrm{Tr}(f_*)=1 \neq 0$ and so the map has a fixed point.

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Using the observation from the other (now deleted) answer that the image of $S^2$ is closed, we deduce that $S^2 \setminus f(S^2)$ is open and connected, so choosing $x$ outside of the image, and a small neighborhood $U$ around it, we see that $f$ maps $S^2 \setminus U \to S^2 \setminus U$, where you can apply the Brower fixed point theorem.