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Rational homology and Quillen's conjecture
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Abstract
We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup $O_p(G)$, then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.
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Possibly relevant literature (found by the agent, not checked)
- Kevin I. Piterman, Iván Sadofschi Costa, Antonio Viruel, Acyclic 2-dimensional complexes and Quillen's conjecture (arXiv:1907.02141, 2019; rev. 2020). Proves Quillen's conjecture for groups of p-rank 3 via G-invariant acyclic 2-dimensional complexes, including cases outside Aschbacher–Smith; this known case is not covered by the manuscript's account of prior results or its bibliography.
- Kevin Iván Piterman, Spherical p-group complexes arising from finite groups of Lie type (arXiv:2403.07489, 2024). Studies p-group complexes of groups with a self-centralising Lie-type normal subgroup in characteristic p and their nonzero Euler characteristic, which bears on the defining-characteristic components the manuscript handles via Piterman (arXiv:2607.27500).
- Kevin Ivan Piterman, On the frame complex of symplectic spaces (arXiv:2305.02940, 2023). Computes homology vanishing for the frame complex of a symplectic space over a finite field, which is related to the symplectic frame cycles constructed in the manuscript's Section 7.
- Kevin Ivan Piterman, John Shareshian, Volkmar Welker, Posets of decompositions in spherical buildings (arXiv:2511.22531, 2025). Studies decomposition and common-bases posets in spherical buildings, related to the manuscript's frame/apartment-chain construction, which cites only the unitary case (Piterman–Welker 2024).
- Elias Gabriel Minian, Kevin Ivan Piterman, The fundamental group of the p-subgroup complex (arXiv:1903.03549, 2019). Studies the homotopy type (fundamental group) of the same p-subgroup complexes, giving background on their topology beyond rational homology.
Digests and expositions
- Latent Seminar, Quillen homology: frame cycles and a coefficient that survives (2026-10-07). Explains the manuscript's frame-cycle dimension bounds, fixed-complement restriction and homology propagation, explicitly limiting its review to selected mechanisms rather than independent verification of the complete proof. [family-level]
- Gradient Damage, Kervaire Invariant Problem at Prime 3, The Grothendieck Homotopy Hypothesis and 8 More (video compilation, 2026-10-15). Chapter 5, beginning at 1:24, introduces Quillen's subgroup-poset conjecture and the manuscript's claimed rational-homology conclusion as a visual illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 310
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Rational-homology-and-Quillens-conjecture-September-24-2026,
author = {{OpenAI}},
title = {{Rational homology and Quillen's conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf}{OAI:Rational-homology-and-Quillens-conjecture-September-24-2026}},
year = {2026}
}Family 310: Quillen's conjecture in rational homology
Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.
This is the only paper in this family.