Homologies of moduli space $\mathcal{M}_{2,1}$

TL;DR

This paper analyzes the topological structure of the moduli space of genus 2 curves with one marked point, using chord diagrams to describe its cell structure and compute Betti numbers.

Contribution

It introduces a novel approach using chord diagrams to describe the cell structure and adjacency of the moduli space, enabling Betti number calculations.

Findings

Cell structure of $\, ext{M}_{2,1}$ described

Boundary operator matrices constructed

Betti numbers computed over $\, ext{Q}$

Abstract

We consider the space $\mathcal{M}_{2,1}$ --- the open moduli space of complex curves of genus 2 with one marked point. Using language of chord diagrams we describe the cell structure of $\mathcal{M}_{2,1}$ and cell adjacency. This allows us to construct matrices of boundary operators and compute Betty numbers of $\mathcal{M}_{2,1}$ over $\mathbb{Q}$.