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

We prove a discrete version of the Lusternik–Schnirelmann (L–S) theorem for discrete Morse functions and the recently introduced simplicial L–S 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 L–S category of K.