Magnitude homology equivalence of Euclidean sets

Doña Mateo, Adrián; Leinster, Tom

doi:10.2140/agt.2026.26.599

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Magnitude homology equivalence of Euclidean sets

Adrián Doña Mateo and Tom Leinster

Algebraic & Geometric Topology 26 (2026) 599–624

DOI: 10.2140/agt.2026.26.599

Abstract

Magnitude homology is an $ℝ^{+}$ -graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carathéodory.

Keywords

magnitude, magnitude homology, metric space

Mathematical Subject Classification

Primary: 18G90

Secondary: 51F99, 52A99

References

Publication

Received: 14 July 2024

Revised: 14 February 2025

Accepted: 17 March 2025

Published: 11 February 2026

Authors

Adrián Doña Mateo
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Tom Leinster
School of Mathematics University of Edinburgh Edinburgh United Kingdom

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2026, the article will become Open Access in early 2026. Otherwise, this article (and all 2026 articles) will be available only to paid subscribers.

Volume 26, issue 2 (2026)