# The Koszul property as a topological invariant and a measure of singularities

## TL;DR

This paper demonstrates that the Koszul property of a certain algebra associated with a cell complex is a topological invariant, linking algebraic properties to topological features and singularities.

## Contribution

It proves that the Koszul property of R(X) is a topological invariant and establishes conditions on local singularities affecting this property.

## Key findings

- Koszul property is a topological invariant for cell complexes.
- Conditions on local singularities influence the Koszul property.
- R(X) being Koszul relates to the topology of the cell complex.

## Abstract

Cassidy, Phan and Shelton associate to any regular cell complex X a quadratic K-algebra R(X). They give a combinatorial solution to the question of when this algebra is Koszul. The algebra R(X) is a combinatorial invariant but not a topological invariant. We show that nevertheless, the property that R(x) be Koszul is a topological invariant. In the process we establish some conditions on the types of local singular- ities that can occur in cell complexes X such that R(X) is Koszul, and more generally in cell complexes that are pure and connected by codimension one faces.