A finitely generated counterexample to the Eilenberg–Ganea conjecture

Comments and discussion

• Grigori Avramidi, Comment #243 on “The Mathocalypse” (Shtetl-Optimized, 2026-10-10). Identifies result 249 as the longstanding Bestvina–Brady candidate and describes its small SU(2)-holonomy argument as an alternative to proving the epsilon-map conjecture or Whitehead’s conjecture.
• Grigori Avramidi, Comment on “The Curve Bends You” (Substack, 2026-10-08). Discusses the finitely generated Bestvina–Brady example and suggests investigating other, finitely presented candidates.
• Xiaolei Wu, quoted in 100+ reactions to 100+ solutions (Proofs and Prompts, 2026-10-08). In a short reaction to the group theory section of the release, Wu writes: "Eilenberg–Ganea is false—not so surprising, perhaps, but how?", and distinguishes the result from the still-open Whitehead conjecture.

Formalization

• OpenAI math release, A finitely generated Eilenberg–Ganea counterexample: Lean scope notes (release Lean docs, 2026-10-06). Describes the formalization as constructing a finitely generated, residually finite group with integral cohomological dimension two that has a three-dimensional classifying space but no two-dimensional one, and links the comparator statement EilenbergGanea.lean. [family-level]
• OpenAI math release, ComparatorChallenges/EilenbergGanea.lean (openai/math, October 2026). This is the comparator statement source_main: the height kernel of the Artin group on seedGraph is finitely generated and residually finite, has cohomologicalDimension = 2, and has a 3-dimensional but no 2-dimensional HasClassifyingSpace, with no bound on the number of cells.
• franzhusch, Eilenberg-Ganea Conjecture (issue #1794) (google-deepmind/formal-conjectures, opened 2026-01-20). An open request, written before the release, to add the conjecture's statement; it is labelled "needs-prerequisites" because Mathlib lacks group cohomological dimension and K(G,1) complexes, and it has no comments about this paper.

Possibly relevant literature (found by the agent, not checked)

• Elton Pasku, An answer to the Whitehead asphericity question (arXiv:2107.12293, v1 2021-07-26, v2 2026-10-05). Claims a positive answer to Whitehead's asphericity question; under the Bestvina–Brady alternative cited in the paper's history section (BB1997, Theorem 8.7), such an answer would bear on the same Eilenberg–Ganea/Whitehead dichotomy.
• Noel Brady, Ian Leary, Brita Nucinkis, On algebraic and geometric dimensions for groups with torsion (J. London Math. Soc. 64(2), 489–500, 2001). Gives counterexamples to the proper-action (Bredon) analogue of the Eilenberg–Ganea conjecture, the variant that the paper's introduction explicitly sets aside ("no family of subgroups or proper-action variant is involved").
• Ian J. Leary, Nansen Petrosyan, On dimensions of groups with cocompact classifying spaces for proper actions (arXiv:1504.02704, 2015; revised 2017). Gives groups whose virtual and Bredon cohomological dimensions are both two but which have no 2-dimensional contractible proper G-CW-complex; this is background for the Coxeter candidates that the introduction compares with Bestvina–Brady groups via LP2026.
• Laurent Bartholdi and Roman Mikhailov, Finite Chains of Two-Complexes and Acyclic Covers (arXiv:2610.10074, 2026-10-07). This AI-assisted work on acyclic regular covers and finite chains of two-complexes, including an SL(2,5) presentation complex, offers a comparison for the manuscript’s acyclic seed and its discussion of Whitehead’s asphericity problem.

Digests and expositions

• AI Proved What?, Dimension two, or three? An AI preprint claims a counterexample to the Eilenberg–Ganea conjecture (Episode 249, October 2026). Provides a short video and transcript explaining the claimed dimension gap, the Bestvina–Brady construction and the small quaternion-labeling obstruction. [AI-generated per source]
• Open Mathematics Courses, A finitely generated counterexample to the Eilenberg–Ganea conjecture (undated). Offers eleven lessons with proofs and exercises covering the manuscript’s argument, explicitly described as written and self-checked by Claude Opus 5.5 with independent review incomplete. [AI-generated per source]