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

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