Why does there exist a continuous map with no fixed point $f\colon S^n\to S^n$ when $n\ge 1$?
I can find a continuous map that has no fixed points for the case $n=1$ but I fail to see how this generalizes.
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Take $f$ to be the antipodal map.
answered Dec 3, 2015 at 19:12
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$\begingroup$ does the map $f(x)=-x$ work? also can you explain why there are no fixed points if $f$ is antipodal map? $\endgroup$ Dec 4, 2015 at 7:32
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$\begingroup$ That's exactly the antipodal map. It can't have fixed point because the only point such that $x=-x$ is $0$, which does not belong to any $S^n$. $\endgroup$ Dec 4, 2015 at 19:28