Morse theory on graphs

TL;DR

This paper develops Morse theoretical techniques to analyze the equivariant cohomology ring of a graph with a torus action, providing a combinatorial approach to understanding its structure.

Contribution

It introduces a graphical Morse theory framework to study the equivariant cohomology of graphs with torus actions, extending classical Morse theory to a combinatorial setting.

Findings

Established a Morse-theoretic approach to graph cohomology

Derived structural properties of the equivariant cohomology ring

Provided tools for computing cohomology in graph-based models

Abstract

Let $\Gamma$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $\Gamma$ is defined by a map, $\alpha$, which assigns to each oriented edge e of $\Gamma$ a one-dimensional representation of G (or, alternatively, a weight, $\alpha_e$, in the weight lattice of G). For the assignment, $e \to \alpha_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(\Gamma, \alpha)$ an equivariant cohomology ring, $H_G(\Gamma)=H(\Gamma,\alpha)$. By definition this ring contains the equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, as a subring, and in this paper we will use graphical versions of standard Morse theoretical techniques to analyze the structure of $H_G(\Gamma)$ as an $\SS(\fg^*)$-module.