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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 $f(f(x))=x$?

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Yes. Have you learned the Lefschetz fixed point theorem yet? –  David Speyer Dec 12 '12 at 19:56
    
No, give me some reference please @DavidSpeyer –  Tomás Dec 12 '12 at 20:42
    
Have you learned what cohomology is? If so, en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem isn't bad. –  David Speyer Dec 12 '12 at 21:03
    
Is there a more analytic way to solve this problem? Because i got it in a course of dynamical systems @DavidSpeyer –  Tomás Dec 12 '12 at 21:05
    
Not that I see. But I'm sure there are other approaches in the world. I'll leave it there. –  David Speyer Dec 12 '12 at 21:40

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