The Eilenberg–Ganea conjecture
From OpenAI Aftermath
The Eilenberg–Ganea conjecture asks whether every group of integral cohomological dimension two has a two-dimensional classifying space. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026, claims a counterexample. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The integral cohomological dimension of a discrete group is the projective dimension of the trivial module Z over its group ring, and its geometric dimension is the least dimension of a classifying space; the first never exceeds the second. Eilenberg and Ganea proved in 1957 that the two agree when the cohomological dimension is at least three. In dimension one, Stallings proved that a finitely generated group of cohomological dimension one is free, and Swan removed the finite-generation hypothesis. The remaining case is the Eilenberg–Ganea conjecture in dimension two.
- Bestvina and Brady used kernels of the height homomorphism of right-angled Artin groups to separate homological from homotopical finiteness. For suitable flag triangulations of a spine of the Poincaré homology sphere they proved that at least one of the Eilenberg–Ganea conjecture and Whitehead's asphericity conjecture fails. Dicks and Leary described such kernels by edge generators and relators, and Howie studied related examples through plus constructions. Leary and Petrosyan recently compared Coxeter and Bestvina–Brady candidates under a conjecture.
The original result
- Claim. Theorem 1.1 of A finitely generated counterexample to the Eilenberg–Ganea conjecture states that there is a finitely generated residually finite group G with integral cohomological dimension two and geometric dimension three. One may take G to be the kernel of the height homomorphism of the right-angled Artin group of a finite acyclic flag triangulation of the presentation complex of the group with generators x, y and relations x2 = y5 and x2 = (xy−1)3. No two-dimensional classifying space for G exists, whatever the number of its cells. The manuscript states that G has a length-two resolution by finitely generated free modules, so it is of type FL and FP∞, and that it is not finitely presented.
- Method, as described: G acts freely on a two-dimensional acyclic level complex whose triangular boundary chains form a free basis of the cycle module. A hypothetical two-dimensional classifying space is compared with this basis through an aligned presentation; a degree argument for word equations in SU(2) on finite quotients and a compactness argument then show that every sufficiently small edge labelling by unit quaternions that is trivial around the triangles would be trivial around every closed path. A transport construction from a nontrivial representation of the fundamental group of the flag complex into SU(2) gives small labellings that are trivial around the triangles but not around some closed path.
- Source (openai/math repository, commit
7ef5c0c): A finitely generated counterexample to the Eilenberg–Ganea conjecture, dated 23 September 2026. - Formalization, as described by the repository. The scope note lean/docs/249.md states that the formalization constructs a finitely generated residually finite group of cohomological dimension two that has a three-dimensional classifying space but no two-dimensional one, and links the comparator statement EilenbergGanea.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 28 October 2026) |
|---|---|---|---|---|
| 2026-10-10 | G. Avramidi, comment 243 on the post The Mathocalypse of the blog Shtetl-Optimized | Remark (blog comment) | Identifies the group of the manuscript with the Bestvina–Brady group that had been thought to be a counterexample for decades, and describes the proof as an argument with small SU(2) holonomy, instead of a proof of Bestvina and Brady's epsilon-map conjecture or of Whitehead's conjecture, which the comment describes as surviving for now. | Blog comment of 10 October 2026. Read on 28 October 2026. |
| 2026-10-15 | A finitely generated counterexample to the Eilenberg-Ganea conjecture, a course on Open Mathematics Courses, source repository kokunoyumeto/open-math-courses-public | Reading (exposition) | Eleven lessons with proofs, examples and solved exercises: cohomological and geometric dimension, word maps on unit quaternions and degree, aligned presentations of the cycle module, finite quotients and the comparison theorem, right-angled Artin groups and residual finiteness, the cube complex and the acyclic level complex, an acyclic seed with quaternion monodromy, and small transport and the counterexample. | The course states that it was written and self-checked by Claude Opus 5.5 of Anthropic and that independent review is not complete. It first appears in the course repository on 15 October 2026. Read on 28 October 2026. |
| 2026-10-17 | AI Proved What?, episode 249 Dimension two, or three?, remade version of 22 October 2026 | Reading (exposition) | Video explainer of the claim of a finitely generated, residually finite group of cohomological dimension two and geometric dimension three, a Bestvina–Brady group built from a flag triangulation of the complex with two loops and two discs for the relations of the manuscript. The episode states that a finitely presented example, anything about Whitehead's conjecture, and the versions with families of subgroups are not claimed. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 19 October 2026: posted as a preprint: OpenAI, 25 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; formalized in Lean: yes, for the exact complex. The record also notes that Wikipedia edits mentioning the claim were reverted pending independent sources and that no error has been reported. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
