Lusternik–Schnirelmann category of simplicial complexes and finite spaces

Abstract

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 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 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 LScategory are obtained in the setting of finite topological spaces.

Simplicial Lusternik-Schnirelmann category

“…In a previous paper [8] we introduced the so-called simplicial Lusternik-Schnirelmann category of a simplicial complex. This invariant, denoted by scat K, is defined directly from the combinatorial structure of the complex K, instead of considering the topological LS-category of the geometric realization |K|, by replacing the notion of homotopy by that of contiguity.…”

Discrete topological complexity

“…One of Farber's main results for topological complexity relates it to a well known classical invariant, the Lusternik-Schnirelmann category [1]. In this section we get analogous results for the discrete setting, by using the simplicial LS-category of a simplicial complex introduced by the authors in [5,6].…”

“…It is common in algebraic topology to consider a normalized version of concepts such asŠvarc genus, topological complexity and LScategory cat X is often used, as in [1], in such a way that contractible spaces have category zero. This is the convention we followed in our papers [5,6] and we will maintain it here. However, sometimes a non-normalized definition (which is equivalent to cat X + 1) can be used in some papers, as Farber did in [3].…”

Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

“…Now let g be the discrete Morse function given in Figure 3.4. Now we consider the pairs (9,9), (8,8), (7, 7), (5, 5), (3, 3) ∈ V g and obtain corresponding values l 9 = 10, l 8 = 8, l 7 = 7, l 5 = 6 and l 3 = 3. The corresponding strong collapses are then given by .…”

“…A version of this result can be stated as follows: There are many "smooth" versions of this theorem in various contexts (see for example [13]). The aim of this paper is to view Forman's discrete Morse theory from a different perspective in order to prove a discrete version of the L-S theorem compatible with the recently defined simplicial L-S category developed by three of the authors [8]. This simplicial version of L-S category is suitable for simplicial complexes.…”