Magnitude homology equivalence of Euclidean sets
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.
Fields of Study & Subjects
Topological and Geometric Data Analysis
Publication details
- Journal / source
- Algebraic & Geometric Topology
- Volume
- 26
- Issue
- 2
- Pages
- 599–624
- Publication date
- 11 February 2026
- Language
- EN