Free groups not good?
Three AI preprints claim counterexamples to four-dimensional disc embedding and to the finitely presented form of Wall’s question
Covers result 305 in OpenAI’s catalog: all 3 papers. See its connections to other results and the work it cites.
4:54. English captions available. The video loads from YouTube when you press play.
Chapters
The claims, in plain words
In dimension five and up, the Whitney trick says that if the intersection points of two submanifolds of complementary dimension cancel in pairs algebraically, a disc can push each pair apart. That trick drives surgery and the h-cobordism theorem. In dimension four the Whitney disc can itself hit the surfaces. Freedman rescued the trick in 1982, for topological discs, when the fundamental group is trivial, and work of Freedman and Quinn, Freedman and Teichner, and Krushkal and Quinn extended it to so-called good groups, including finite, solvable and subexponential-growth groups. The central question became whether free groups are good. In 1983 Freedman proposed, with cautious neutrality, that the Whitehead double of the Borromean rings is not slice, which would make the disc theorem fail for free groups; later experts called the question open. Separately, in 1979, Wall asked whether every Poincaré duality group is the fundamental group of a closed aspherical manifold; Davis answered no in 1998, with groups that are not finitely presented.
Result 305 is three OpenAI papers, 133 pages, dated September 24, 2026, and made public October 6. Two of them independently claim a counterexample to the unrestricted disc embedding conjecture: a compact smooth four-manifold with discs and framed dual spheres meeting every algebraic condition, whose boundary circles still bound no disjoint locally flat discs (in the second paper, none respecting the boundary framings). One of them deduces, through a theorem of Powell, Ray and Teichner, that the free group on two letters is not good, which, if correct, goes the way Freedman guessed; so unrestricted topological surgery and the five-dimensional s-cobordism theorem cannot both hold. The third paper, which builds on one of the first two, claims a finitely presented Poincaré duality group of dimension four, with a finite classifying space, that is not the fundamental group of any closed aspherical topological four-manifold.
What they don’t claim
Which of surgery or s-cobordism fails; anything that settles Freedman’s named test cases, such as the Whitehead double of the Borromean rings; anything about homology manifolds; or anything in higher dimensions.
Where it stands
- Posted as preprints
- OpenAI, three papers, 133 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No formalization for this item
Last checked October 19, 2026. None of the papers has been revised, and there is no Lean formalization for this item. OSVerify has not reviewed it. In the first twelve days we found no public specialist comment and no reported error.
How the proof goes
- Borromean algebra. The construction installs Brunnian triple commutators along grope boundaries, the algebra of the Borromean rings, three circles linked together though no two are linked. Disjoint discs would let you cut the manifold along three-dimensional walls that miss them, where these commutators become relations and force the flat SL₂(ℂ) deformations to obey a cube-zero law.
- Algebra in characteristic ℓ. Chern–Simons potentials turn this into exact algebraic identities, reduced modulo a large prime, and a finite algebraic theorem rules them out once there are at least five walls. The third paper uses Davis’s reflection trick on a chamber whose marked boundary has no filling shaped like a graph.
The numbers in the video
- Ours: a ball of radius 10 holds 118,097 elements of the free group on two letters, against 221 points of the plane lattice.
- Ours: the Borromean rings have pairwise linking numbers 0, 0, 0, and the commutator [x₁,[x₂,x₃]] of length 10 dies when any one of its three letters is erased everywhere.
- Ours, from the first paper’s definitions: about 10¹⁴ discs. Davis’s trick on a right-angled pentagon glues 32 copies into a surface of genus 5.
Read the source
- A Boundary-Only Obstruction to Four-Dimensional Disk Embedding (PDF), 48 pages.
- A Marked Tensor Obstruction to Four-Dimensional Disk Embedding (PDF), 53 pages.
- A PD4 Group without an Aspherical Manifold Model (PDF), 32 pages.
- The collection’s catalog, where this is result 305.
The 3 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
In dimension 5 and up, the Whitney trick says that if the intersection points of two submanifolds of complementary dimension cancel in pairs algebraically, a disc can push each pair apart. That trick drives surgery and the h-cobordism theorem. In dimension 4 the Whitney disc can itself hit the surfaces. Our computer counts the bookkeeping: 1,000 points that cancel algebraically need one Whitney disc per pair, five hundred in all, each embedded and disjoint from everything. Freedman rescued the trick in 1982, for topological discs, when the fundamental group is trivial; one consequence is a closed 4-manifold with intersection form E8, signature 8, that cannot be smoothed. Work of Freedman and Quinn, Freedman and Teichner, and Krushkal and Quinn extended it to so-called good groups, including finite, solvable and subexponential-growth groups. The central question became whether free groups are good. Our computer compares growth: a ball of radius 10 holds 118,097 elements of the free group on two letters, against 221 points of the plane lattice. In 1983 Freedman proposed, with cautious neutrality, that the Whitehead double of the Borromean rings is not slice, which would make the disc theorem fail for free groups; later experts called the question open. Separately, in 1979, Wall asked whether every Poincaré duality group is the fundamental group of a closed aspherical manifold. Davis answered no in 1998, with groups that are not finitely presented. The finitely presented form stayed open, including in dimension 4.
1:42The claims
Result 305 is three OpenAI papers, 133 pages, dated September 24 and made public October 6, 2026. Two of them independently claim a counterexample to the unrestricted disc embedding conjecture: a compact smooth 4-manifold with discs and framed dual spheres meeting every algebraic condition, whose boundary circles still bound no disjoint locally flat discs, in one version, or none respecting the boundary framings, in the other. One of them deduces, through a theorem of Powell, Ray and Teichner, that the free group on two letters is not good, which, if correct, goes the way Freedman guessed. It follows that unrestricted topological surgery and the 5-dimensional s-cobordism theorem cannot both hold; the paper does not say which fails. The third paper, which builds on one of the first two, claims a finitely presented Poincaré duality group of dimension 4, with a finite classifying space, that is not the fundamental group of any closed aspherical topological 4-manifold: a negative answer to the finitely presented form of Wall’s question in dimension 4. Not claimed: which of surgery or s-cobordism fails, anything about homology manifolds, or higher dimensions.
3:03How the proof goes
The construction installs Brunnian triple commutators along grope boundaries, the algebra of the Borromean rings, three circles linked together though no two are linked. Our computer checks the model: all three pairwise linking numbers are zero, and the commutator of length 10 dies as soon as any one of its three letters is erased everywhere. Disjoint discs would let you cut the manifold along three-dimensional walls that miss them, where these commutators become relations and force the flat SL₂(ℂ) deformations to obey a cube-zero law. Chern–Simons potentials turn this into exact algebraic identities, reduced modulo a large prime, and a finite algebraic theorem rules them out once there are at least 5 walls. The construction is huge: by our count from the first paper’s definitions, about 10¹⁴ discs. The third paper uses Davis’s reflection trick on a chamber whose marked boundary has no filling shaped like a graph; our computer shows the trick on a right-angled pentagon, where 32 copies glue into a surface of genus 5.
4:12Where it stands
All three papers are unrefereed and unchanged since release, and there is no Lean formalization. In the first 12 days we found no public specialist comment and no reported error. The papers also do not settle the specific test cases Freedman named, such as the Whitehead double of the Borromean rings. If they hold, the four-dimensional disc embedding theorem genuinely needs a restriction on the fundamental group: nonabelian free groups are not good, and in dimension 4 a finitely presented Poincaré duality group need not be the fundamental group of a closed aspherical manifold.
Spot a mistake?
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