Comparing combinatorial models of moduli space and their compactifications

TL;DR

This paper compares two combinatorial models of the moduli space of 2D cobordisms, establishing a homotopy equivalence and showing their natural compactifications are homeomorphic.

Contribution

It provides an explicit homotopy equivalence between Bödigheimer's radial slit configurations and Godin's admissible fat graphs, including their compactifications.

Findings

Homotopy equivalence between the two models.

Cellular homeomorphism of their compactifications.

Explicit construction of the 'critical graph' map.

Abstract

We compare two combinatorial models for the moduli space of two-dimensional cobordisms: B\"odigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmonic compactification and Sullivan diagrams respectively, and prove that the homotopy equivalence induces a cellular homeomorphism between these compactifications.