Reversing orientation homeomorphisms of surfaces

TL;DR

The paper proves that certain orientation-reversing diffeomorphisms of surfaces, which preserve a Morse function and its level set components, have order two up to isotopy, extending classical symmetry results.

Contribution

It establishes that such orientation-reversing diffeomorphisms squared are isotopic to the identity, generalizing known plane symmetry properties to surfaces with Morse functions.

Findings

Reversing orientation diffeomorphisms squared are isotopic to identity.

Results extend to maps with isolated singularities on surfaces.

Applicable to a broader class of maps beyond Morse functions.

Abstract

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. is a mirror symmetry with respect to some line. The obtained results hold in fact for a larger class of maps with isolated singularities from connected compact orientable surfaces to the real line and the circle.