Let $f:S^{1}\to S^{1}$ be a non-trivial homeomorphism. Suppose we have $f\circ f = id$. Can we say that $f$ has to be orientation reversing?
I think this is true. But I couldn't come up with a proof.
Thanks in advance for any kind of help!
2026-03-25 07:16:36.1774422996
Is an order two homeomorphism of $S^{1}$ orientation reversing?
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This is wrong and there are infinitely many counter examples, eg any conjugate ($ gfg^{-1} $) of the $180º $-rotation by any homeomorphism $ g $ ($ g $ itself does not have to respect the orientation).
What is interesting now is that the result becomes true (and not so hard to prove) if you replace the circle by the real line! In particular it is true on the circle if you ask additionally that the homeomorphism has a fixed point.