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A boundary-only obstruction to four-dimensional disk embedding

OpenAI · September 24, 2026

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Abstract

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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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

  • Vyacheslav Krushkal, 4-manifold topology and collision rank of groups (arXiv:2609.25198, 2026-09-21). Introduces "collision rank" and proves that groups of finite collision rank are good, including groups that are not subexponentially amenable; relevant to the paper's description of the known class of good groups (Section 1.1) next to its claim that $F_2$ is not good (Corollary 1.2).

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Cite

@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}
}

Family 305: 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.

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