Higher discrete homotopy groups of graphs

Bob Lutz

doi:10.5802/alco.151

Abstract

This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if G is a graph containing no 3- or 4-cycles, then the nth discrete homotopy group A n (G) is trivial for all n≥2. Second we exhibit for each n≥1 a natural homomorphism ψ:A n (G)→ℋ n (G), where ℋ n (G) is the nth discrete cubical singular homology group, and an infinite family of graphs G for which ℋ n (G) is nontrivial and ψ is surjective. It follows that for each n≥1 there are graphs G for which A n (G) is nontrivial.

Highlights

In [6] a new homotopy theory for simplicial complexes was introduced, motivated by a search for qualitative invariants in the study of complex systems and their dynamics [7]
This paper studies the problem of computing higher discrete homotopy groups
We prove a similar theorem for discrete homotopy

Summary

Introduction

In [6] a new homotopy theory for simplicial complexes was introduced, motivated by a search for qualitative invariants in the study of complex systems and their dynamics [7]. Aqn(K, σ0), n 1, called the discrete homotopy groups of K. In contrast to classical homotopy theory, the groups (1) are defined combinatorially. For n = 1, the group (1) is called the discrete fundamental group, and is well understood. Aq1(K, σ0) ∼= π1(X, x0), where π1(X, x0) is the classical fundamental group for some x0 ∈ X Results like these have enabled computations of Aq1(K, σ0) for interesting simplicial complexes, including Coxeter complexes of finite Coxeter groups [8, 9]. The basic tools for computing classical higher homotopy groups, already a difficult problem in general, have no known discrete analogs. In [1] a higher-dimensional version of (2) was obtained, assuming a “plausible” cubical analog of the simplicial approximation theorem. This paper studies the problem of computing higher discrete homotopy groups. Discrete singular cubical homology, A-theory, Hurewicz theorem

Basic definitions

Triangle- and square-free graphs

Cones and suspensions

Toward a discrete Hurewicz theorem

Final remarks