Morse Theory for Complexes of Groups

TL;DR

This paper develops an equivariant discrete Morse theory framework for simplicial complexes with group actions, enabling the construction of Morse complexes of groups that preserve equivariant homotopy types.

Contribution

It introduces a 2-categorical criterion for compatible acyclic matchings and constructs Morse complexes of groups that recover the original complex up to equivariant homotopy.

Findings

Morse complexes of groups are homotopy equivalent to the original complex.

The method generalizes discrete Morse theory to equivariant settings.

Provides a new tool for studying group actions on simplicial complexes.

Abstract

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.