The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a homeomorphism $f$ of $S^1$ has a rational rotation number $r$ ($r=\lim_{n\to\infty} \frac{F^n(x)-x}{n}$ where $F$ is a "lift" of $f$ to $\mathbb{R}$) and if $f$ is "orientation-preserving", then every periodic point of $f$ has the same period. My question is simply:
What does it mean (rigorously) for a homeomorphism $f$ of the circle $S^1$ to be "orientation-preserving"?
Thank you in advance!
Best Answer
Let $p:\Bbb{R} \longrightarrow S^1$ be the fundamental covering map $t \mapsto e^{2\pi i t}$. Then $f$ preserves orientation if there exists some increasing homeomorphism $g: \Bbb{R} \longrightarrow \Bbb{R}$ such that $p \circ g = f \circ p$, i.e. if $f$ can be lifted to an increasing homeomorphism of $\Bbb{R}$.
Note that any homeomorphism of $\Bbb{R}$ is necessarily monotone, so this gives the notion of "preserving / swapping the orientation" for a homeomorphism of $S^1$.