Minimal projective resolution and magnitude homology of geodetic metric spaces

TL;DR

This paper develops a method to compute the magnitude homology of geodetic metric spaces using minimal projective resolutions, providing explicit calculations and characterizations for various graphs.

Contribution

It introduces a minimal projective resolution approach to compute magnitude homology and characterizes when finite geodetic spaces are diagonal, with explicit examples.

Findings

Magnitude homology of geodetic spaces is a free -module.

Finite geodetic spaces are diagonal iff they contain no 4-cuts.

Explicit computations for cycle, Petersen, Hoffman-Singleton, and Moore graphs.

Abstract

Asao-Ivanov showed that magnitude homology is 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 $\mathbb Z$-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 and Gu.