Simplicial Lusternik-Schnirelmann category

TL;DR

This paper introduces a new combinatorial invariant called the simplicial LS-category for finite simplicial complexes, extending classical topological concepts to a purely combinatorial setting.

Contribution

It defines the simplicial LS-category as a strong homotopy invariant, generalizing graph arboricity, and develops algebraic topology tools in this combinatorial framework.

Findings

The invariant is well-defined for finite simplicial complexes.

It generalizes the classical notion of arboricity.

It enables algebraic topology techniques in combinatorial settings.

Abstract

The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms, that generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that allows to develop all the machinery of algebraic topology which is costumary in the classical theory of Lusternik-Schnirelmann category.