Matching and Independence Complexes Related to Small Grids

TL;DR

This paper studies the topology of independence complexes related to small grid graphs, providing new discrete Morse matchings, bounds on homology, and calculating Euler characteristics.

Contribution

It introduces a discrete Morse matching for a family of independence complexes including grid matching complexes, advancing understanding of their topological properties.

Findings

Determined the dimensions of chain spaces for Morse complexes.

Established bounds on the location of non-trivial homology groups.

Calculated the Euler characteristic of the independence complexes.

Abstract

The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.