# Four-dimensional disk embedding and Wall's conjecture > Claims the two-generator free group is not good for 4D surgery, plus a counterexample to Wall's question. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-305/ Last updated: 2026-10-14 **Claims the two-generator free group is not good for 4D surgery, plus a counterexample to Wall's question.** **Trust:** Claimed, not checked · **Papers:** 3 · **First paper:** September 24, 2026 · **Field:** Topology · **Entry:** 305 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. In four dimensions, topologists build embedded discs from algebraic intersection data. Freedman and Quinn showed this works when the fundamental group is good. The open test case was the free group on two generators. These papers claim the disc construction fails without the good-group condition, so that free group is not good. A third paper uses this to claim a four-dimensional duality group with no aspherical manifold model. ## At a glance - **First step:** 1982 (Michael Freedman, embedded discs in simply connected four-manifolds through Casson handles) - **Significance:** Landmark. If the proofs hold, they settle the free-group case of the disc embedding problem, a central open question in four-manifold topology. - **Trust:** Claimed, not checked - **Field:** Topology (3 papers) ## The explanation Two surfaces in a four-dimensional space can cross at points. Each crossing gets a sign, +1 or −1. Suppose two crossings have opposite signs. Their algebraic count is +1 + (−1) = 0, but they still meet at 2 points. The Whitney trick slides one surface along a disc to remove such a pair. The disc is called a Whitney disc. In five or more dimensions this works. In four dimensions the Whitney disc can itself hit the surfaces. Casson replaced the disc by an infinite tower of discs, called a Casson handle. Freedman showed it behaves like an ordinary thickened disc. This settled simply connected four-manifolds, where every loop can shrink to a point. Freedman and Quinn extended the method to good groups. The fundamental group records the loops in a space up to deformation. A group is good if certain loops in grope models can be shrunk. A grope is a tower of surfaces glued along curves. The test models have free fundamental groups, so the free group is the central case[1]. Count words in the free group on a and b, where no letter sits next to its inverse. There are 4 × 3⁴ = 324 such words of length 5. In the grid group ℤ², only 20 elements need exactly 5 steps. Freedman and Teichner showed that groups of slow, subexponential growth are good[2]. The free group grows exponentially, and its case remained open[3]. The disc embedding conjecture says the discs always exist, with no condition on the group. The first paper claims to disprove it[4]. Its discs come with dual spheres, which meet each disc once in the algebraic count. Even with no framing or dual sphere asked for, no disjoint discs exist[5]. It concludes that the free group on two generators is not good[6]. A second paper claims a separate disproof, in which the new discs must keep the boundary framings[7]. It studies flat connections with values in the matrix group SL₂[8]. These are rules for moving data around loops without twisting. An algebraic obstruction rules out the needed data when at least five pieces are active[9]. The third paper takes up Wall's question. A Poincaré duality group has the algebra of a closed manifold's fundamental group. Wall asked whether each one is the fundamental group of a closed aspherical manifold[10]. A space is aspherical if its universal cover can shrink to a point. Davis gave examples that are not finitely presented[11]. For finitely presented groups, a 2026 survey records no known counterexample[12]. The third paper claims a finitely presented four-dimensional example with a finite classifying space. It claims no closed aspherical topological four-manifold has that group[13]. It builds on the second paper's obstruction[14]. It refutes the all-dimensions question, not each dimension[15]. How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The arguments combine four-manifold geometry with long algebraic and analytic steps. No specialist review of these papers has been published. **What it doesn't show.** This does not give a counterexample in every dimension, and it does not settle which other groups are good. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Landmark if the proofs hold, since the free-group case was a central open problem. The arguments are long, with no Lean check and no published specialist review. ## Words used - **Whitney trick:** A move that removes a pair of opposite-sign crossings by sliding a surface across a disc. - **Fundamental group:** The group of loops in a space from a base point, counted up to deformation. - **Good group:** A group for which Freedman and Quinn's disc construction is known to work. - **Free group:** The group of words in letters and their inverses, with no rules beyond cancelling neighbours. - **Dual sphere:** A sphere that meets one chosen disc once in the signed count and the others zero times. - **Aspherical:** A space whose universal cover can be shrunk to a point. - **Poincaré duality group:** A group whose algebra matches that of the fundamental group of a closed aspherical manifold. - **Classifying space:** An aspherical space with the given group as its fundamental group. ## The original papers (3) - [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf) - September 24, 2026 - [A marked tensor obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf) - September 24, 2026 - [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf) - September 24, 2026 ## Quoted lines from the paper (15) [1] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > The thickenings used in this test have free fundamental groups, which explains the central role of the free-group case. [2] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > Freedman--Teichner extended the positive theory to all groups of subexponential growth. [3] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > The nonabelian free-group case remained open in the 2025 account; see also for its place among four-manifold problems. [4] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. [5] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > Theorem … disproves it by excluding every disjoint locally flat filling of the specified boundary circles, even when no output framing or dual sphere is required. [6] From [A boundary-only obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > Consequently, the free group on two generators is not good in the sense of Freedman--Quinn, and neither is any group containing it as a subgroup. [7] From [A marked tensor obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > We disprove the unrestricted four-dimensional disk-embedding conjecture. [8] From [A marked tensor obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > For each edge \(Y_i\), we study based flat \(SL_2\)-connections with one torus holonomy fixed. [9] From [A marked tensor obstruction to four-dimensional disk embedding](https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf): > Theorem … rules out precisely these algebraic data when at least five edges are active. [10] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > Wall asked whether every integral Poincar\'e duality group is the fundamental group of a closed aspherical manifold. [11] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > Davis used reflection groups to turn the finiteness examples of Bestvina and Brady into integral $\PD_n$ groups that are not finitely presented, for every $n\geq4$. [12] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > Kielak's 2026 survey records that no counterexamples to this formulation were known. [13] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > We construct a finitely presented integral Poincar\'e duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. [14] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > Our starting point is the marked geometric and tensor obstruction in \emph{A marked tensor obstruction to four-dimensional disk embedding}. [15] From [A PD4 group without an aspherical manifold model](https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf): > Theorem … refutes the assertion quantified simultaneously over all dimensions $n\geq3$, without asserting a counterexample in every dimension. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [The Hilbert–Smith conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-304/) - Topology - [The purely cosmetic surgery conjecture](https://closedform.nishachay.tech/e/openai-math-2026-306/) - Topology - [Quillen's conjecture in rational homology](https://closedform.nishachay.tech/e/openai-math-2026-310/) - Topology - [The Hovey–Strickland and Chai conjectures](https://closedform.nishachay.tech/e/openai-math-2026-311/) - Topology - [The four-dimensional Singer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-315/) - Topology - [Topology](https://closedform.nishachay.tech/science/mathematics/topology/) - Field · 18 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Open since the 1980s](https://closedform.nishachay.tech/open-since/1980s/) - Era · 60 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20305%5D%20)