Dimension two, or three?
An AI preprint claims a counterexample to the Eilenberg–Ganea conjecture
Covers result 249 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:34. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
A classifying space for a group is a space whose loops form the group and whose universal cover is contractible: a circle for the integers, a torus for pairs of integers. The smallest possible dimension of one is the group’s geometric dimension. Algebra gives a second number, the cohomological dimension: the length of the shortest projective resolution of the integers over the group ring, which never exceeds the first. In 1957 Eilenberg and Ganea showed the two agree from three up, and Stallings and Swan settled dimension one. That left dimension two: does every group of cohomological dimension two have a two-dimensional classifying space? In 1997 Bestvina and Brady built candidates, and showed that for those built from the Poincaré homology sphere, at least one fails: this conjecture, or Whitehead’s asphericity conjecture.
Result 249 is one OpenAI preprint, 25 pages, dated September 23, 2026. It claims a finitely generated, residually finite group of cohomological dimension two and geometric dimension three, which would disprove the Eilenberg–Ganea conjecture: the group has no two-dimensional classifying space, however many cells one allows. The group is a Bestvina–Brady group built from a flag triangulation of the complex with two loops, x and y, and two discs for x² = y⁵ and x² = (xy⁻¹)³.
What it doesn’t claim
A finitely presented example (this group is not finitely presented); anything about Whitehead’s conjecture, which stays open; or the versions with families of subgroups, which were already known to fail.
Where it stands
- Posted as a preprint
- OpenAI, 25 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Yes, for the exact complex (docs)
Last checked October 19, 2026. The preprint has not been revised. In Lean, the theorem is formalized for this exact complex: about 34,000 lines with no unproved steps. We did not build it, and it is missing from the repository’s formalization list. OSVerify has not reviewed the paper. A commenter on Scott Aaronson’s blog said it is the Bestvina–Brady group long thought to be a counterexample. Wikipedia edits mentioning the claim were reverted pending independent sources, and no error has been reported.
How the proof goes
- The test. Put a tiny rotation, a unit quaternion near 1, on every edge of the two-dimensional slice the group acts on, so that the product around each triangle is 1. Commuting labels would give 1 around every loop, because the homology vanishes. With quaternions, a loop could still twist.
- If a two-dimensional classifying space existed. It could be of any size; the paper turns it into a presentation lined up with the triangles. The first-order error cancels exactly, leaving only commutators. A degree count on products of copies of SU(2), going back to Gerstenhaber and Rothaus, solves the remaining equations. Finite quotients and compactness then force every small enough flat labelling to give 1 around every loop.
- But labels that twist exist. A direction map sends each element of the Artin group to a point of the seed complex and changes slowly. Pulling back the 120-element quaternion representation gives tiny labels, flat on every triangle, with a loop that twists. So no two-dimensional classifying space exists, while a three-dimensional one does.
The numbers in the video
- Ours: the seed complex has zero reduced homology, yet its fundamental group has 120 elements. The paper’s map sends them onto 120 unit quaternions, the vertices of the 600-cell, and the complex’s second homotopy group is 119 copies of the integers.
- Ours: the Lean version’s flag complex has 103 vertices, 372 edges and 270 triangles, zero homology and a fundamental group of order 120.
- Ours: the direction map’s step bound held in 14,854 random steps and was reached exactly. Signed solution counts of small word maps on SU(2) (3, −2, 3, −4, 2) matched their determinants numerically.
Read the source
- A finitely generated counterexample to the Eilenberg–Ganea conjecture (PDF), 25 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 249.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
How many dimensions does a group need? The integers are the loops around a circle; pairs of integers, the loops on a torus. A space whose loops form the group, and whose universal cover is contractible, is a classifying space; the smallest dimension of one is the group’s geometric dimension. Algebra gives a second number, the cohomological dimension: the length of the shortest projective resolution of the integers over the group ring, which can never exceed the first. In 1957 Eilenberg and Ganea showed the two agree from three up; Stallings and Swan settled dimension one. That left dimension two: does every group of cohomological dimension two have a 2-dimensional classifying space? In 1997 Bestvina and Brady built candidates, and showed that for those built from the Poincaré homology sphere, at least one conjecture fails: this one, or Whitehead’s asphericity conjecture. The trouble is that homology can be blind. Our computer took the complex with two loops, x and y, and two discs, for the relations x² = y⁵ and x² = (xy⁻¹)³. Its reduced homology is zero, yet its fundamental group has 120 elements, sent onto 120 unit quaternions, the corners of a 4-dimensional solid called the 600-cell; and the complex’s second homotopy group is 119 copies of the integers.
1:28The claim
Result 249 is one OpenAI preprint, 25 pages, dated September 23, 2026. It claims a finitely generated, residually finite group of cohomological dimension two and geometric dimension three, which would disprove the Eilenberg–Ganea conjecture: no 2-dimensional classifying space, regardless of how many cells it has. The group is a Bestvina–Brady group built from a flag triangulation of that complex. Not claimed: a finitely presented example, since this group is not finitely presented; anything about Whitehead’s conjecture, which stays open; or the versions with families of subgroups, which were already known to fail. Our computer rebuilt the Lean version’s complex: 103 vertices, 372 edges and 270 triangles, homology zero, and a fundamental group of order 120.
2:25How the proof goes
The test: put a tiny rotation, a unit quaternion near one, on every edge of the 2-dimensional slice the group acts on, so that the product around each triangle is one. With commuting labels, every loop would give one, because the homology vanishes. With quaternions, a loop could still twist. Suppose a 2-dimensional classifying space existed, of any size. The paper turns it into a presentation lined up with the triangles; the first-order error cancels exactly, leaving only commutators. A degree count on products of copies of SU(2), going back to Gerstenhaber and Rothaus, where the degree equals a determinant, solves the remaining equations; our numerical checks gave signed counts 3, −2, 3, −4 and 2, matching. Finite quotients and compactness then force every small enough flat labelling to give one around every loop. But the paper builds small labels that twist. A direction map sends each element of the Artin group to a point of the seed complex, changing by at most 2 divided by the word length per step; in our 14,854 random steps the bound was never exceeded, and it was reached exactly. Pulling back the quaternion representation gives tiny labels, flat on every triangle, with a loop that twists. So no 2-dimensional classifying space exists, while a 3-dimensional one does.
3:52Where it stands
The preprint is unrefereed and unchanged since release. In Lean, the theorem is formalized for this exact complex, about 34,000 lines with no unproved steps; we did not build it, and it is missing from the repository’s formalization list. OSVerify has not reviewed it. A commenter on Scott Aaronson’s blog said it is the Bestvina–Brady group long thought to be a counterexample; Wikipedia edits were reverted, and no error has been reported. If it holds, algebra can be fooled: a group that looks 2-dimensional to algebra needs three dimensions to be built, 69 years after Eilenberg and Ganea.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.