  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 $\mathbb{Q}$.
