# Manifold Matching Complexes

## TL;DR

This paper characterizes when the matching complex of a graph forms a homology manifold, revealing that, except in dimension two, these manifolds are primarily spheres or balls, providing a complete classification.

## Contribution

It provides a complete characterization of graph-matching complex pairs where the complex is a homology manifold, including cases with or without boundary.

## Key findings

- Most matching complexes are spheres or balls in higher dimensions.
- The paper classifies all pairs where the matching complex is a homology manifold.
- Dimension two is an exception in the classification.

## Abstract

The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper we completely characterize the pairs (graph, matching complex) for which the matching complex is a homology manifold, with or without boundary. Except in dimension two, all of these manifolds are sphere or balls.