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Magnitude homology of geodesic space

Kiyonori Gomi

TL;DR

This work connects magnitude homology to the geometry of geodesic spaces by giving concrete, geometrically meaningful descriptions of low-degree magnitude homology. It shows that H^_2 can be read from simplicial complexes A(a,b) and B^(a,b), while H^_3 in geodesic spaces is governed by H_1(A(a,b)); a non-branching assumption yields a complete diagonal description of all H^_n(X) in terms of sequences of geodesics and reveals a torsion-free structure. The authors introduce an invariant ν_f for third-magnitude classes that captures intersection properties of geodesic-driven edge paths, and demonstrate nontrivial examples. The results include explicit characterizations for classic spaces like S^1 and S^d, establish degeneracy in the higher spectral sequence under non-branching, and lay groundwork for further links between magnitude homology and geodesic geometry.

Abstract

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then, on a geodesic metric space, we interpret the description by means of geodesics. The third magnitude homology of a geodesic metric space also admits a description in terms of a simplicial complex. Under an assumption on a metric space, the simplicial description allows us to introduce an invariant of third magnitude homology classes as an intersection number. Finally, we provide a complete description of all the magnitude homology groups of a geodesic metric space which fulfils a certain non-branching assumption.

Magnitude homology of geodesic space

TL;DR

This work connects magnitude homology to the geometry of geodesic spaces by giving concrete, geometrically meaningful descriptions of low-degree magnitude homology. It shows that H^_2 can be read from simplicial complexes A(a,b) and B^(a,b), while H^_3 in geodesic spaces is governed by H_1(A(a,b)); a non-branching assumption yields a complete diagonal description of all H^_n(X) in terms of sequences of geodesics and reveals a torsion-free structure. The authors introduce an invariant ν_f for third-magnitude classes that captures intersection properties of geodesic-driven edge paths, and demonstrate nontrivial examples. The results include explicit characterizations for classic spaces like S^1 and S^d, establish degeneracy in the higher spectral sequence under non-branching, and lay groundwork for further links between magnitude homology and geodesic geometry.

Abstract

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then, on a geodesic metric space, we interpret the description by means of geodesics. The third magnitude homology of a geodesic metric space also admits a description in terms of a simplicial complex. Under an assumption on a metric space, the simplicial description allows us to introduce an invariant of third magnitude homology classes as an intersection number. Finally, we provide a complete description of all the magnitude homology groups of a geodesic metric space which fulfils a certain non-branching assumption.

Paper Structure

This paper contains 16 sections, 50 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Magnitude homology
  4. A simplicial complex in magnitude homology
  5. Smoothness spectral sequence
  6. Geodesic metric space
  7. The second magnitude homology
  8. The case of general metric space
  9. The case of geodesic metric space
  10. The third magnitude homology
  11. The case of geodesic metric space
  12. An invariant of the third magnitude homology
  13. Higher magnitude homology
  14. A vanishing result about $E^2_{p, 1}$
  15. A cycle associated to geodesics
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(X, d)$ be a metric space, and $\ell$ a real number. For any $a, b \in X$, we have where $H_0(B^\ell(a, b))$ is the zeroth homology group of a simplicial complex $B^\ell(a, b)$.

Figures (1)

  • Figure 1: The vertices and the edges of a cube

Theorems & Definitions (105)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 95 more