Fixed point free homeomorphisms of the complex plane

TL;DR

This paper proves that the set of fixed point free homeomorphisms of the complex plane forms a dense G_delta subset within the group of all homeomorphisms, highlighting their topological and algebraic properties.

Contribution

It establishes that fixed point free homeomorphisms constitute a conjugacy invariant dense G_delta subset in the homeomorphism group of the complex plane.

Findings

Fixed point free homeomorphisms form a dense G_delta set.

The group of homeomorphisms is a metric group under uniform convergence.

Fixed point free homeomorphisms are conjugacy invariant.

Abstract

Our purpose in this article is to prove that the group $H({\bf C})$ of homeomorphisms of the complex plane ${\bf C}$ is a metric group equipped with the metric induced by uniform convergence of homeomorphisms and their inverses on compacts and the set $$\left\{ h \in H({\bf C}) : ( \forall z \in {\bf C} )( h(z) \neq z ) \right\} $$ of fixed point free homeomorphisms of the complex plane is a conjugacy invariant dense $G_{\delta }$ subset of $H({\bf C})$.