Magnitude homology of metric spaces and order complexes

TL;DR

This paper links the magnitude homology of metric spaces to order complexes of interval posets, providing new computational tools and applications including vanishing results and torsion detection.

Contribution

It introduces a novel description of magnitude homology in terms of order complexes and applies this to derive new properties and examples.

Findings

Higher magnitude homology vanishes for convex Euclidean subsets

Magnitude homology encodes hole diameters

Constructed a graph with torsion in third magnitude homology

Abstract

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.