The Structure and Singularities of Arc Complexes

TL;DR

This paper classifies when arc complexes of surfaces with boundary are spherical, revealing their structure and singularities, which has implications for understanding the topology of Riemann's moduli space.

Contribution

It provides a classification of spherical arc complexes for surfaces with boundary, extending classical results and connecting to moduli space singularities.

Findings

Identifies conditions for arc complexes to be spherical

Describes the singularities of the cellular compactification

Connects arc complex topology to Riemann's moduli space

Abstract

A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The main result of this paper is the determination of those arc complexes Arc(F) that are also spherical. This classification has consequences for Riemann's moduli space via its known identification with an analogous arc complex in the punctured case with no boundary. Namely, the essential singulities of the natural cellular compactification can be described.