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The Eilenberg-Ganea conjecture

For a group GG, the cohomological dimension cd G\mathrm{cd}\,G is the projective dimension of the trivial Z[G]\mathbb Z[G]-module Z\mathbb Z, and the geometric dimension gd G\mathrm{gd}\,G is the least dimension of a K(G,1)K(G,1) CW complex; always cd G≤gd G\mathrm{cd}\,G\le\mathrm{gd}\,G. Eilenberg and Ganea (1957) proved cd G=gd G\mathrm{cd}\,G=\mathrm{gd}\,G whenever cd G≥3\mathrm{cd}\,G\ge3, and Stallings and Swan settled dimension one. Bestvina and Brady (1997) showed that at least one of the Eilenberg-Ganea conjecture and Whitehead's asphericity conjecture is false, without deciding which. Does every group of cohomological dimension two admit a two-dimensional classifying space?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Geometric group theory; cohomological and geometric dimension of groups
Posed by
Samuel Eilenberg and Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. of Math. 65 (1957)
Year posed
1957
Years open
69y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a finitely generated residually finite group GG with cd G=2\mathrm{cd}\,G=2 and gd G=3\mathrm{gd}\,G=3; no two-dimensional K(G,1)K(G,1) exists regardless of the number of cells. GG is a Bestvina-Brady group (the kernel of the height map of a right-angled Artin group); it is of type FL and FP∞_\infty but not finitely presented. The obstruction compares an acyclic two-dimensional level complex with any hypothetical aspherical presentation, using small SU(2) labelings and a degree argument. It does not give a finitely presented counterexample, and by itself it does not decide Whitehead's asphericity conjecture.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture as the paper states it (with the 1957 reference); it claims a finitely generated, residually finite group GG, the height kernel of a right-angled Artin group over an acyclic flag triangulation of the presentation complex of ⟨x,y∣x2=y5,x2=(xy−1)3⟩\langle x,y\mid x^2=y^5,x^2=(xy^{-1})^3\rangle, with cd G=2\mathrm{cd}\,G=2 and gd G=3\mathrm{gd}\,G=3. The proof was not refereed. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/249.md, and its solution module OAI.Topology.EilenbergGanea.Main exists at the pinned commit. The statement ComparatorChallenges/EilenbergGanea.lean was read here: it defines the group as the kernel of the height map on the Artin group of an explicit seed graph and asserts it is finitely generated, residually finite, has cohomological dimension 2 (projective dimension of the trivial module), has a 3-dimensional classifying CW space, and has no 2-dimensional one in any universe, with no bound on cells. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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