2025/03/31 by Gregory Lupton, Lupton, Gregory, Nicholas A. Scoville +3
Computer Science · Mathematics · #(Primary) 55U10 05E45 #(Secondary) 55M99 55Q99 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2503.23651
openalex publication_date 2025/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the ordinary (topological) fundamental group of the spatial realization. We define a counterpart to the edge group that likewise gives a combinatorial version of the second (higher) homotopy group. Working entirely combinatorially, we show our group is an abelian group and also respects products. We show that our combinatorially defined group is isomorphic to the ordinary (topological) second homotopy group of the spatial realization.