Tessellations of Moduli Spaces and the Mosaic Operad

TL;DR

This paper introduces a new operad based on mosaics formed by polygons with marked diagonals, relating to moduli spaces of punctured spheres and their tiling by associahedra, with connections to braid group operads.

Contribution

It constructs a novel cyclic operad of mosaics, describing moduli space points via combinatorial blow-ups and linking fundamental groups to braid group operads.

Findings

Spaces are tiled by associahedra

Fundamental groups form an operad similar to braid groups

Operad structure derived from combinatorial polygon configurations

Abstract

We construct a new (cyclic) operad of `mosaics' defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the sets of real points of the moduli space of punctured Riemann spheres, which are naturally tiled by Stasheff associahedra. We (combinatorially) describe them as iterated blow-ups and show that their fundamental groups form an operad with similarities to the operad of braid groups.