Topological aspects of the dynamical moduli space of rational maps

TL;DR

This paper explores the topological structure of the space of rational maps on the Riemann sphere, revealing its rational acyclicity, fundamental group, and higher homotopy groups, and demonstrating it is not nilpotent.

Contribution

It provides new insights into the topology of the moduli space of rational maps, including computations of fundamental and higher homotopy groups, extending previous knowledge.

Findings

The space is rationally acyclic.

The fundamental group is computed.

Higher homotopy groups are characterized.

Abstract

We investigate the topology of the space of M\"obius conjugacy classes of degree $d$ rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks of some higher homotopy groups of the parameter space of degree $d$ rational maps allowing us to extend the previously known range. Moreover, we show that this parameter space is not nilpotent.