Discrete Morse theory for symmetric Delta-complexes

TL;DR

This paper extends discrete Morse theory to symmetric Delta-complexes and applies it to prove the contractibility of a key subcomplex in the moduli space of tropical abelian varieties.

Contribution

It generalizes Forman's discrete Morse theory to symmetric Delta-complexes and demonstrates its application in tropical geometry.

Findings

Proved the contractibility of the coloop subcomplex in the tropical moduli space.

Extended discrete Morse theory to symmetric Delta-complexes.

Provided new tools for analyzing the topology of tropical moduli spaces.

Abstract

We generalize Forman's discrete Morse theory to the context of symmetric $\Delta$-complexes. As an application, we prove that the coloop subcomplex of the link of the origin $LA^{\mathrm{trop},\mathrm{P}}_g$ in the moduli space of principally polarized tropical abelian varieties of dimension $g$ with respect to the perfect cone decomposition is contractible.