Fixed point free homeomorphisms of the complex plane

Abstract

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