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Torsion in Magnitude homology theories

Patrick Martin, Radmila Sazdanovic

TL;DR

The paper investigates torsion phenomena in magnitude homology theories for finite graphs, focusing on magnitude and Eulerian magnitude homology. It extends prior torsion results by proving that any finitely generated abelian group can occur as a torsion subgroup of Eulerian magnitude homology, using two independent proofs: a Kaneta–Yoshinaga embedding and an Asao–Izumihara geometric construction. It further develops torsion-free and p-torsion constructions without relying on manifolds by exploiting regular CW structures and Moore spaces, and provides explicit computations for trees, complete graphs, and star trees. The work clarifies the relationships among MH, EMH, and discriminant DMH, and identifies diagonality phenomena and invariance properties, with potential applications in graph theory and combinatorial topology.

Abstract

In this article, we analyze the structure and relationships between magnitude homology and Eulerian magnitude homology of finite graphs. Building on the work of Kaneta and Yoshinaga, Sazdanovic and Summers, and Asao and Izumihara, we provide two proofs of the existence of torsion in Eulerian magnitude homology, offer insights into the types and orders of torsion, and present explicit computations for various classes of graphs.

Torsion in Magnitude homology theories

TL;DR

The paper investigates torsion phenomena in magnitude homology theories for finite graphs, focusing on magnitude and Eulerian magnitude homology. It extends prior torsion results by proving that any finitely generated abelian group can occur as a torsion subgroup of Eulerian magnitude homology, using two independent proofs: a Kaneta–Yoshinaga embedding and an Asao–Izumihara geometric construction. It further develops torsion-free and p-torsion constructions without relying on manifolds by exploiting regular CW structures and Moore spaces, and provides explicit computations for trees, complete graphs, and star trees. The work clarifies the relationships among MH, EMH, and discriminant DMH, and identifies diagonality phenomena and invariance properties, with potential applications in graph theory and combinatorial topology.

Abstract

In this article, we analyze the structure and relationships between magnitude homology and Eulerian magnitude homology of finite graphs. Building on the work of Kaneta and Yoshinaga, Sazdanovic and Summers, and Asao and Izumihara, we provide two proofs of the existence of torsion in Eulerian magnitude homology, offer insights into the types and orders of torsion, and present explicit computations for various classes of graphs.

Paper Structure

This paper contains 8 sections, 42 theorems, 64 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Torsion in Eulerian magnitude homology
  4. Torsion via Asao-Izumihara construction
  5. Torsion without manifolds
  6. Regular CW structures and torsion
  7. Magnitude Homology: computations and relations
  8. Conjectures and future directions

Key Result

Proposition 2.3

Magnitude $\#G$ of any graph $G$ is equal to

Figures (8)

  • Figure 1: The Kenta-Yoshinaga construction for $\mathbb{RP}^2$
  • Figure 2: A graph $G$
  • Figure 3: The poset $P_4^\sigma$
  • Figure 4: The poset $P_6^\tau$
  • Figure 5: The bijection $\varphi:\mathcal{I}_4^\sigma\to AC_4(3)$
  • ...and 3 more figures

Theorems & Definitions (92)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: Lemma 11 hepworth2015categorifying
  • Definition 2.7
  • Proposition 2.8: Theorem 8 hepworth2015categorifying
  • Definition 2.9
  • Definition 2.10
  • ...and 82 more