On the Homology of Certain Smooth Covers of Moduli Spaces of Algebraic Curves

TL;DR

This paper introduces a method to compute the homology of smooth covers of moduli spaces of algebraic curves with level structures, applying it to low genus cases and confirming known topological invariants.

Contribution

A new general method for homology computation of smooth covers of moduli spaces using stratification lifting, validated for genus up to 2.

Findings

Computed Betti numbers for genus 1 and 2 covers

Results align with Penner and Harer-Zagier Euler characteristics

Method provides a framework for higher genus cases

Abstract

We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We apply this method for $g\leq 2$ and obtain Betti numbers; these results are consistent with Penner and Harer-Zagier results on Euler characteristics.