Strong discrete Morse theory and simplicial L-S category: A discrete version of the Lusternik-Schnirelmann Theorem

TL;DR

This paper develops a discrete analogue of the Lusternik-Schnirelmann theorem using discrete Morse theory and simplicial categories, introducing new critical objects and establishing bounds on topological complexity.

Contribution

It introduces a new notion of critical objects for discrete Morse functions and proves a discrete Lusternik-Schnirelmann theorem relating these objects to simplicial categories.

Findings

Critical objects guarantee strong homotopy equivalence between sublevel complexes.

Number of critical objects bounds the simplicial Lusternik-Schnirelmann category.

Established a discrete version of the classical Lusternik-Schnirelmann theorem.

Abstract

We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian's sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical objects of a discrete Morse function defined on $K$ is an upper bound for the non-normalized simplicial Lusternik-Schnirelmann category of $K$.