Lusternik-Schnirelmann category of simplicial complexes and finite spaces

TL;DR

This paper introduces a homotopy-invariant Lusternik-Schnirelmann category for simplicial complexes based on contiguity, and explores its properties and implications for finite topological spaces.

Contribution

It defines a new simplicial category invariant using contiguity, proves its homotopy invariance, and links it to finite topological spaces through poset relations.

Findings

Category is homotopy invariant under strong equivalences.

Maximum category value is attained in the core of the complex.

New results connect simplicial and finite topological space categories.

Abstract

In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of category are obtained in the setting of finite topological spaces.