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Field independence of the first seven Betti numbers of flag complexes
Abstract
In 2006, Katzman showed that the first six Betti numbers of the Stanley--Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let τ(d) be the least number of vertices of a flag complex whose d-th reduced integral homology has torsion. We prove that τ(d)≥ d+10 for every d≥0. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph G, the first seven Betti numbers of the edge ideal I(G) are field independent.
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BibTeX
@misc{javadekar2026field,
title = {{Field independence of the first seven Betti numbers of flag complexes}},
author = {Omkar Javadekar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28376},
eprint = {2609.28376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28376},
}
RIS
TY - UNPB
TI - Field independence of the first seven Betti numbers of flag complexes
AU - Omkar Javadekar
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28376
UR - https://arxiv.org/abs/2609.28376
AB - In 2006, Katzman showed that the first six Betti numbers of the Stanley--Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let τ(d) be the least number of vertices of a flag complex whose d-th reduced integral homology has torsion. We prove that τ(d)≥ d+10 for every d≥0. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph G, the first seven Betti numbers of the edge ideal I(G) are field independent.
ER -