Cellular Decomposition of Compactified Hurwitz Spaces

Summary

We describe a cellular decomposition of compactified Hurwitz spaces, generalizing the cellular decomposition of moduli spaces of punctured Riemann surfaces M_g,n. The main motivation for this work is the integration of cohomology classes on compactified Hurwitz spaces, and is provided by Witten's conjecture on moduli spaces of Riemann surfaces with spin, and by the fact (proved in this paper) that these spaces are closely related to Hurwitz spaces of Galois cyclic coverings. This article also aims to give all details of the Harer-Kontsevich theorem.

1. Introduction

In the last ten years, the geometry of moduli spaces of punctured Riemann surfaces have seen an increasing interest and known some striking progress [20] in connection with physic's theories [26]. This also concerns some generalizations of moduli spaces of punctured surfaces, like moduli spaces of stable maps, or moduli spaces of Riemann surfaces with spin [27] [17]. In this paper, we firstly show that the topological framework which have allowed M. Kontsevich to compute in a combinatorial way some Chern classes on M_g,n (a key point of his proof of Witten's conjecture [26]) extends to the setting of Hurwitz spaces, and secondly we sketch an analogy between Hurwitz spaces of cyclic coverings and moduli spaces of Riemann surfaces with spin. In a forthcoming work, our results will be used to study the cohomology of Hurwitz spaces for cyclic coverings.

The first one is to work with this equivalence relation. This is the approach of W. J. Harvey [11] [15], and A. Kuribayashi [22] in the setting of Teichimiller theory.