Group Actions on Spaces of Rational Functions

Abstract

Let ${\mathrm{Hol}}_d$ be the space consisting of all holomorphic maps $f:S^2\to S^2$ of degree $d$. The group ${\mathrm{Hol}}_1={\mathrm{PSL}}_2(C)$ acts on ${\mathrm{Hol}}_d$ freely by the post-composition and we shall study the orbit space $X_d={\mathrm{Hol}}_1\backslash {\mathrm{Hol}}_d$. As an application, we shall determine the homotopy types of the universal covering spaces of ${\mathrm{Hol}}_d$ and $X_d$ explicitly.