Magnitude homology of tope graphs

Junnosuke Koizumi

arXiv:2607.11863v1 Primary: math.CO Secondary: math.AT

Abstract

We completely determine the magnitude homology of tope graphs of real hyperplane arrangements. Their ranks can be described as the Hilbert functions of the Stanley--Reisner rings of certain simplicial complexes naturally associated with the arrangements. For Coxeter arrangements, this gives a computation of the magnitude homology of the Cayley graph of the corresponding Coxeter group. We also prove the homological reciprocity for central arrangements conjectured by Koizumi--Liu. The proof combines poset combinatorics, the Edelman--Walker theorem, and Alexander duality.

Disclosure

“ncluding even cycles and the Cayley graph of Sn . Use of AI. In writing this paper, we used AI in the following ways. • Some ideas used in the proofs of Lemma 4.1, Lemma 5.12, Lemma 5.13, and Proposition 5.14 were suggested by ChatGPT-5.5 Pro. • We used ChatGPT-5.5 Pro to improve the prose and to identify grammatical and mathematical errors. All AI-generated suggestions and text incorporated into the paper were carefully checked by the author. The author tak”
PDF page 3

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T18:51:52.331900Z

Structural counts

Pages 26 pdf

Theorems 5 source

Lemmas 15 source

Propositions 7 source

Corollaries 5 source

Definitions 7 source

Displayed equations 162 source

Bibliography entries 72 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file MH.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.