Lusternik–Schnirelmann category of simplicial complexes and finite spaces

Abstract

In this paper we establish a natural definition of Lusternik–Schnirelmann category for simplicial complexes via the well known notion of contiguity. This simplicial category has the property of being invariant under strong equivalences, and it 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 simplicial geometric category. We prove that the maximum value over the simplicial 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 LS-category are obtained in the setting of finite topological spaces.