A triangulation of a homotopy-Deligne-Mumford compactification of the Moduli of curves

TL;DR

This paper constructs a triangulation of a compactification of the moduli space of punctured surfaces, closely related to the Deligne-Mumford compactification, preserving homotopy equivalence and providing a new combinatorial perspective.

Contribution

It introduces a new triangulation of a moduli space compactification that is homotopy equivalent to the Deligne-Mumford compactification, with a surjective map having contractible fibers.

Findings

The triangulation is closely related to the Deligne-Mumford compactification.

The constructed compactification is homotopy equivalent to the Deligne-Mumford space.

A surjective map with contractible inverse images links the new and existing compactifications.

Abstract

We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image of each point is contractible. In particular our compactification is homotopy equivalent to the Deligne-Mumford compactification.