The Face Group of a Simplicial Complex
Algebraic Topology
2025-05-23 v2 Combinatorics
Abstract
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.
Cite
@article{arxiv.2503.23651,
title = {The Face Group of a Simplicial Complex},
author = {Gregory Lupton and Nicholas A. Scoville and P. Christopher Staecker},
journal= {arXiv preprint arXiv:2503.23651},
year = {2025}
}
Comments
37 pages, 14 figures. New version acknowledges work of Grandis