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 $f(f(x))=x$?
Asked
Active
Viewed 210 times
4
-
Yes. Have you learned the Lefschetz fixed point theorem yet? – David E 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, http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem isn't bad. – David E 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 E Speyer Dec 12 '12 at 21:40