Fixed point sets of isotopies on surfaces

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 2-sphere that preserve orientation and pointwise fix some given closed connected set $F$.