Result 305, Topology

Four-dimensional disk embedding and Wall's conjecture

The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

Disproof or counterexample

The bigger picture

Why it matters

In four-dimensional topology, algebraic evidence that crossing disks should be separable may not guarantee actual separation. The manuscripts claim counterexamples that also expose a gap between groups resembling manifold symmetries and groups realizable by manifolds.

What changes?

The boundary-only manuscript reports a compact, oriented, smooth four-manifold containing finitely many disk maps, which may cross themselves or each other. They have framed algebraic dual spheres: auxiliary spheres with untwisted normal directions and prescribed algebraic intersection counts. Nevertheless, their boundary circles cannot bound pairwise disjoint, locally flat disks, meaning disks that look standard near every point. The failure persists without prescribing the replacements' homotopy classes or normal framings. This contradicts disk embedding without a restriction on the fundamental group, which records loops.

What does that help mathematicians do?

The claimed obstruction rules out the free group on two generators, and every group containing it, as "good" in the Freedman–Quinn sense, limiting the groups for which disk-embedding guarantees hold. A companion manuscript reports a finitely presented integral Poincaré duality group of dimension four with a finite classifying space, but no realization as the fundamental group of a closed aspherical topological four-manifold. Thus these finiteness and duality properties are insufficient for manifold realization.

Are there practical applications?

The immediate value is foundational: these claims identify limits of translating algebraic data into four-dimensional geometry. Researchers would need additional hypotheses before using intersection counts to obtain embedded disks, or group duality to obtain manifold models. The latter distinction concerns aspherical manifolds, whose universal covering spaces contract to a point, showing why algebraically plausible candidates cannot automatically be treated as geometric examples.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

A boundary-only obstruction to four-dimensional disk embedding

September 24, 2026 48 pages

We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. 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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026,
  author = {{OpenAI}},
  title = {{A boundary-only obstruction to four-dimensional disk embedding}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf}{OAI:A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026}},
  year = {2026}
}

A marked tensor obstruction to four-dimensional disk embedding

September 24, 2026 53 pages

We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026,
  author = {{OpenAI}},
  title = {{A marked tensor obstruction to four-dimensional disk embedding}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf}{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026}},
  year = {2026}
}

A PD4 group without an aspherical manifold model

September 24, 2026 32 pages

We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-PD4-group-without-an-aspherical-manifold-model-September-24-2026,
  author = {{OpenAI}},
  title = {{A PD4 group without an aspherical manifold model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf}{OAI:A-PD4-group-without-an-aspherical-manifold-model-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.