Tessellations of moduli spaces and the mosaic operad. (English) Zbl 0968.32009

Devadoss, Satyan L.

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 \(\overline{{\mathcal M}_0^n}(\mathbb R)\) of a certain compactification of the moduli space of Riemann spheres with \(n\) punctures \({\mathcal M}_0^n(\mathbb C)\). 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 \(\overline{{\mathcal M}_0^n}(\mathbb R)\) can be tesselated by \(1/2\cdot(n-1)!\) associahedra of dimension \(n-3\). This gives a formula for the Euler characteristic of \(\overline{{\mathcal M}_0^n}(\mathbb R)\). The combinatorics of associahedra is further used to investigate the relationship by blow-ups between \(\overline{{\mathcal M}_0^n}(\mathbb R)\) and the projective space PG\(_{n-3}\mathbb R\).
For the entire collection see [Zbl 0924.00035].

Reviewer: Michael Joswig (Berlin)

MSC:

32G15

Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

moduli space; configuration space; complex projective line; associahedron