The rational homology of independence complexes of Kneser graphs on triples
Abstract
The independence complex of the Kneser graph KG(n,k) is the simplicial complex of pairwise intersecting families of n-subsets of [2n+k]; equivalently, it is a Vietoris--Rips complex of the n-subsets of [2n+k] under the symmetric-difference metric. Feng and Wang asked whether, for n=3, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every k≥0, the only nonzero reduced Betti numbers using rationals of Ind(KG(3,k)) are β6=binomk+57+29binomk+67 and β9=binomk+66. The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer--Vietoris spectral sequence which reduces the problem to the six-element sets.
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BibTeX
@misc{feng2026rational,
title = {{The rational homology of independence complexes of Kneser graphs on triples}},
author = {Ziqin Feng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17394},
eprint = {2610.17394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17394},
}
RIS
TY - UNPB
TI - The rational homology of independence complexes of Kneser graphs on triples
AU - Ziqin Feng
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17394
UR - https://arxiv.org/abs/2610.17394
AB - The independence complex of the Kneser graph KG(n,k) is the simplicial complex of pairwise intersecting families of n-subsets of [2n+k]; equivalently, it is a Vietoris--Rips complex of the n-subsets of [2n+k] under the symmetric-difference metric. Feng and Wang asked whether, for n=3, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every k≥0, the only nonzero reduced Betti numbers using rationals of Ind(KG(3,k)) are β_6=binomk+57+29binomk+67 and β_9=binomk+66. The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer--Vietoris spectral sequence which reduces the problem to the six-element sets.
ER -