Magnitude homology of geodesic space

TL;DR

This paper explores the structure of magnitude homology groups in geodesic metric spaces, providing descriptions in terms of simplicial complexes and introducing an intersection number invariant under certain conditions.

Contribution

It offers a novel simplicial complex interpretation of magnitude homology in geodesic spaces and characterizes all such groups under a non-branching assumption.

Findings

Second magnitude homology described via zeroth homology of simplicial complexes.

Third magnitude homology admits a simplicial complex description.

Complete classification of magnitude homology groups under non-branching conditions.

Abstract

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then, on a geodesic metric space, we interpret the description by means of geodesics. The third magnitude homology of a geodesic metric space also admits a description in terms of a simplicial complex. Under an assumption on a metric space, the simplicial description allows us to introduce an invariant of third magnitude homology classes as an intersection number. Finally, we provide a complete description of all the magnitude homology groups of a geodesic metric space which fulfils a certain non-branching assumption.