Magnitude homology equivalence of Euclidean sets

Adrián Doña Mateo; Tom Leinster

doi:10.2140/agt.2026.26.599

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arXiv · 2406.11722

Magnitude homology equivalence of Euclidean sets

Adrián Doña Mateo·Tom Leinster

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Abstract

Magnitude homology is an $\mathbf{R}^+$-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.

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BibTeX RIS

Adrián Doña Mateo, Tom Leinster. 2026-02-20. Magnitude homology equivalence of Euclidean sets. https://doi.org/10.2140/agt.2026.26.599

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