Magnitude homology equivalence of Euclidean sets

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.

💡 Research Summary

The paper investigates when two metric spaces have the same magnitude homology, a graded homology theory introduced for enriched categories and later specialized to metric spaces. Rather than the weakest notion (isomorphic homology groups) or the intermediate notion (existence of a map inducing an isomorphism), the authors adopt the strongest notion: two spaces X and Y are magnitude‑homology equivalent if there exist short (1‑Lipschitz) maps f : X→Y and g : Y→X such that the induced maps on Hⁿ,ℓ are mutually inverse for every positive degree n.

The authors first introduce the class of aligned metric spaces. An aligned space satisfies a linear‑interval property: whenever a chain of points x₀≺x₁≺…≺xₙ is given, the interval