Suppose $f:S^2\rightarrow S^2$ is a diffeomorphism. Is true that $f$ must have a fixed point or a point $x\in S^2$ such that $f(f(x))=x$?
Diffeomorphism on the Sphere
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analysis
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0Yes. Have you learned the Lefschetz fixed point theorem yet? – 2012-12-12
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0No, give me some reference please @DavidSpeyer – 2012-12-12
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0Have you learned what cohomology is? If so, http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem isn't bad. – 2012-12-12
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0Is there a more analytic way to solve this problem? Because i got it in a course of dynamical systems @DavidSpeyer – 2012-12-12
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0Not that I see. But I'm sure there are other approaches in the world. I'll leave it there. – 2012-12-12