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Tessellations of moduli spaces and the mosaic operad. (English) Zbl 0968.32009
Meyer, Jean-Pierre (ed.) et al., Homotopy invariant algebraic structures. A conference in honor of J. Michael Boardman. AMS special session on homotopy theory, Baltimore, MD, USA, January 7-10, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 239, 91-114 (1999).
The author studies the geometry and topology of the real points $\overline{{ℳ}_{0}^{n}}\left(ℝ\right)$ of a certain compactification of the moduli space of Riemann spheres with $n$ punctures ${ℳ}_{0}^{n}\left(ℂ\right)$. It is known that the latter can be identified with the configuration space of $n$ distinct points on the complex projective line modulo the action of the group of Möbius transformations. The author proves that $\overline{{ℳ}_{0}^{n}}\left(ℝ\right)$ can be tesselated by $1/2·\left(n-1\right)!$ associahedra of dimension $n-3$. This gives a formula for the Euler characteristic of $\overline{{ℳ}_{0}^{n}}\left(ℝ\right)$. The combinatorics of associahedra is further used to investigate the relationship by blow-ups between $\overline{{ℳ}_{0}^{n}}\left(ℝ\right)$ and the projective space PG${}_{n-3}ℝ$.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory