Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers
Abstract
For a finite group G, let N(G) be the simplicial complex of subsets that do not generate G. We separate a general Frattini-topological reduction from the structure special to finite p-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of G/Φ(G). For a finite p-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of PG(r−1,p), where r=d(G) and q=|Φ(G)|. We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the ℤ-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen--Macaulay classification.
Related
- Realization of permutation modules as homology of CW-complexes
- Homology of matching complexes of 3× n grid graphs
- Homotopy types and torsion in homology of generalized chessboard complexes
- The rational homology of independence complexes of Kneser graphs on triples
- Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals
BibTeX
@misc{garcia2026frattini,
title = {{Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers}},
author = {Luis Alfredo Dupont García},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10296},
eprint = {2610.10296},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10296},
}
RIS
TY - UNPB
TI - Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers
AU - Luis Alfredo Dupont García
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10296
UR - https://arxiv.org/abs/2610.10296
AB - For a finite group G, let N(G) be the simplicial complex of subsets that do not generate G. We separate a general Frattini-topological reduction from the structure special to finite p-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of G/Φ(G). For a finite p-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of PG(r−1,p), where r=d(G) and q=|Φ(G)|. We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the ℤ-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen--Macaulay classification.
ER -