2016/12/28 by D. Fernández-Ternero, Fernández-Ternero, Desamparados, E. Macías–Virgós +5
Computer Science · Mathematics · Medicine · #55M30 #55U05 #57M15 #Algebraic Topology (math.AT) #Clusterin in disease pathology #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1612.08840
openalex publication_date 2016/12/28 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
We prove a discrete version of the Lusternik-Schnirelmann theorem for\ndiscrete Morse functions and the recently introduced simplicial\nLusternik-Schnirelmann category of a simplicial complex. To accomplish this, a\nnew notion of critical object of a discrete Morse function is presented, which\ngeneralizes the usual concept of critical simplex (in the sense of R. Forman).\nWe show that the non-existence of such critical objects guarantees the strong\nhomotopy equivalence (in the Barmak and Minian's sense) between the\ncorresponding sublevel complexes. Finally, we establish that the number of\ncritical objects of a discrete Morse function defined on K is an upper bound\nfor the non-normalized simplicial Lusternik-Schnirelmann category of K.\n