Fixed point sets of isotopies on surfaces

TL;DR

This paper investigates the structure of fixed point sets of surface homeomorphisms, proving the existence of maximal unlinked sets and the connectedness of certain homeomorphism spaces, advancing understanding of surface isotopies.

Contribution

It introduces the concept of maximal unlinked fixed point sets and demonstrates their existence, utilizing Le Calvez' transverse foliations theory.

Findings

Existence of maximal unlinked fixed point sets on surfaces

Connectedness of the space of orientation-preserving homeomorphisms fixing a set

Application of transverse foliations theory to fixed point analysis

Abstract

We consider a self-homeomorphism h of some surface S. A subset F of the fixed point set of h is said to be unlinked if there is an isotopy from the identity to h that fixes every point of F. With Le Calvez' transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the two dimensional sphere that preserves the orientation and pointwise fix some given closed connected set F.