Minimal projective resolution and magnitude homology of geodetic metric spaces
Abstract
In [AI24], magnitude homology is described as a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a consequence, we determine magnitude homology of geodetic metric spaces. We show that it is a free Z-module, and give a recursive algorithm for constructing all cycles. As a corollary, we show that a finite geodetic metric space is diagonal if and only if it contains no 4-cuts. Moreover, we give explicit computations for cycle graphs, Petersen graph, Hoffman-Singleton graph, and a missing Moore graph. It includes another approach to the computation for cycle graphs, which has been studied by Hepworth-Willerton ([HW17]) and Gu ([Gu18]).
Fields of Study & Subjects
Homotopy and Cohomology in Algebraic Topology
Advanced Numerical Analysis Techniques
Fuzzy and Soft Set Theory
Publication details
- Journal / source
- Homology Homotopy and Applications
- Publisher
- International Press of Boston
- Volume
- 28
- Issue
- 2
- Pages
- 123–154
- Publication date
- 1 January 2026
- Language
- EN