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A marked tensor obstruction to four-dimensional disk embedding
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Abstract
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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work in progress
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.
Follow-up works
- OpenAI, Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type (OpenAI Math Release, family 320, 2026-10-04). A later manuscript in the same release claiming a four-dimensional counterexample to the Borel conjecture; it cites this paper (as [MT]) and explicitly imports its algebraicity theorem, together with results of the PD4-group paper of this family. [AI-generated per source]
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 good-group background and the unrestricted disk-embedding conjecture that Theorem 1.1 disproves.
Digests and expositions
- Latent Seminar, Disk embedding and manifold realization: boundary data and tensor obstructions (2026-10-07). This exposition distinguishes all three manuscripts' conclusions and outlines their geometric and algebraic mechanisms, explicitly stating that it is not an independent verification of the complete proofs. [family-level]
Elsewhere
- whataifound: Registry entry
- Emergent Mind: family 305
Independent sites; their content is not reviewed here.
Cite
@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}
}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.
Other papers in this family:
- A boundary-only obstruction to four-dimensional disk embedding · September 24, 2026
- A PD4 group without an aspherical manifold model · September 24, 2026