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

We consider the open moduli space $\mathcal M_{2,1}$ of complex curves of genus 2 with one marked point. Using the 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 $\mathbb Q$.