2023/01/10 by Dominic Desjardins Côté, Côté, Dominic Desjardins
Computer Science · Mathematics · #05C20 #06A06 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #G.2.2 #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2301.03739
openalex publication_date 2023/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a relation on X × Y , we can construct two abstract simplicial complexes called Dowker complexes. The geometric realizations of these simplicial complexes are homotopically equivalent. We show that if two relations are conjugate, then they have homotopically equivalent Dowker complexes. From a self-relation on X , this is a directed graph, and we use the Dowker complexes to study their properties. We show that if two relations are shift equivalent, then, at some power of the relation, their Dowker complexes are homotopically equivalent. Finally, we define a new filtration based on Dowker complexes with different powers of a relation.