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This question already has an answer here:

Prove that every differential application $f:\mathbb{S}^{n}\rightarrow\mathbb{S}^{n}$ such that $deg(f)\neq(-1)^{n+1}$ admits, at least, one fixed point. Here it is worth mentioning that $deg(f)$ denotes the degree of the application $f$ and $\mathbb{S}^{n}$ is the unit sphere in $\mathbb{R}^{n+1}$. Thank you in advance for any help.

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marked as duplicate by MatheinBoulomenos, Krish, Arnaud D., Aqua, Davide Giraudo Nov 23 '17 at 14:17

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Suppose that $f$ has no fixed points. Then I claim that $f$ is homotopic to the antipodal map $i:x\mapsto-x$ on $S^n$. The antipodal map has degree $(-1)^{n+1}$. There is a straight-line homotopy from $f$ to $i$ taking values in $\Bbb R^{n+1}$. As $f$ has no fixed point, this homotopy has values in $\Bbb R^{n+1}-\{O\}$. This homotopy can then be projected on the sphere $S^n$.

In detail this homotopy takes $$(x,t)\to\frac1{|(1-t)f(x)-tx|}((1-t)f(x)-tx).$$

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