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A PD4 group without an aspherical manifold model

OpenAI · September 24, 2026

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Abstract

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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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 [PD]) and explicitly imports its controlled filling and support results, together with the algebraicity theorem of the marked tensor obstruction paper of this family. [AI-generated per source]

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

  • James F. Davis and J. A. Hillman, Aspherical $PD_4$-pairs (arXiv:2608.21717, 2026-08-22). Studies which groups and $PD_3$-complexes are realised by $PD_4$-pairs with aspherical total space; relevant to the paper's relative realization step for its marked chamber pair $(P,S)$, where the bibliography cites only the earlier Davis–Hillman preprint arXiv:2501.12509.

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

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}
}

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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