Simplicial Lusternik-Schnirelmann category

Abstract

The simplicial LS-category of a nite abstract simplicial complex is a new invariant of the strong homotopy type, de ned in purely combinatorial terms. We prove that it generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that it allows to develop many notions and results of alge- braic topology which are costumary in the classical theory of Lusternik{Schnirelmann category. Also we compare the simplicial category of a complex with the LS-category of its geometric realization and we discuss the simplicial analogue of the Whitehead formulation of the LS-category.