Magnitude homology of geodesic spaces

Gomi, Kiyonori

doi:10.2140/agt.2026.26.2377

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Magnitude homology of geodesic spaces

Kiyonori Gomi

Algebraic & Geometric Topology 26 (2026) 2377–2419

DOI: 10.2140/agt.2026.26.2377

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 fulfills a certain nonbranching assumption.

Keywords

magnitude homology, metric space, geodesic metric space

Mathematical Subject Classification

Primary: 18G40, 51F99, 55N35

References

Publication

Received: 29 July 2023

Revised: 29 April 2025

Accepted: 22 July 2025

Published: 1 September 2026

Authors

Kiyonori Gomi
Department of Mathematics Institute of Science Tokyo Tokyo Japan

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Volume 26, issue 7 (2026)