Magnitude homology equivalence of Euclidean sets

TL;DR

This paper characterizes when two Euclidean sets have equivalent magnitude homology by providing a geometric criterion involving inner boundary and core concepts, extending classical convex geometry results.

Contribution

It establishes a concrete geometric condition for magnitude homology equivalence of Euclidean sets, introducing inner boundary and core concepts.

Findings

Provides necessary and sufficient geometric condition for homology equivalence.

Introduces the concepts of inner boundary and core for convex sets.

Strengthens Carathéodory's theorem for closed convex 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 Carath\'eodory.