Topology · Result 305

Four-dimensional disk embedding and Wall's conjecture

3 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 3

A boundary-only obstruction to four-dimensional disk embedding

OpenAI · September 24, 2026

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.

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@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

OpenAI · September 24, 2026

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.

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@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

OpenAI · September 24, 2026

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.

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Cite

@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.