Magnitude homology equivalence of Euclidean sets

TL;DR

The paper addresses when two metric spaces have the same magnitude homology by introducing a concrete geometric framework for Euclidean subsets. It develops the notions of inner boundary $\rho X$ and core $\mathrm{core}(X)$, proving that for nonconvex closed Euclidean sets, magnitude homology equivalence is completely captured by isometry of their cores; nonconvex spaces are shown to be magnitude homology equivalent to their cores via retracts. A key technical route is the restriction to aligned spaces, where magnitude homology is freely generated by thin chains and maps agreeing on $\rho X$ induce chain-homotopic maps in positive degree. The main theorem ties these ideas together, giving several equivalent conditions for magnitude homology equivalence, including the isometry of cores, thereby providing a canonical representative for each equivalence class of nonconvex closed Euclidean sets. This yields a precise geometric criterion for magnitude-homology-based sameness, with implications for understanding geodesic structure through a robust invariant.

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.

Figures (4)

• Figure 1: (a) A closed subset of $\R^2$, with its inner boundary shown in thick blue; (b) the core of the same set, shaded (see Section \ref{['sec:core']}).
• Figure 2: The two cases of (a) Lemma \ref{['lem:boundary_in_interval']}, (b) Lemma \ref{['lem:hull_contained']}, and (c) Proposition \ref{['propn:dichotomy']}. Here $X$ is a closed subset of $E = \R^2$, with inner boundary shown in thick blue. In (b), the subset $A$ of $X$ is the union of the two parallel line segments.
• Figure 3: The proof of Proposition \ref{['prop:core-closure']}.
• Figure 4: Three magnitude homology equivalent spaces, where the set $S$ of Proposition \ref{['propn:remove-interior']} is the union of a disc and a filled square. In (a), $C = \mathop{\mathrm{\overline{conv}}}\nolimits(S)$; in (b), $C$ is a filled ellipse; in (c), $C = \R^2$.

Theorems & Definitions (76)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.5
• proof
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Lemma 2.9
• proof