[#P26] Smooth four-dimensional Poincaré conjecture

Problem. Every smooth closed four-manifold \(M\) that is homotopy equivalent to \(S^4\) is diffeomorphic to the standard smooth four-sphere \(S^4\).

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Definitions and notation

1Context

Dimension four is the remaining exceptional dimension for the smooth generalized Poincaré question.

2Definitions

Definition 1 (A homotopy four-sphere). A homotopy four-sphere is a smooth closed four-manifold with the homotopy type of S^4.

Definition 2 (Diffeomorphic). Diffeomorphic means equivalent by a smooth bijection with a smooth inverse.

3What counts as a solution

Prove that every smooth homotopy four-sphere is diffeomorphic to S^4, or construct a smooth homotopy four-sphere and prove that it is not diffeomorphic to S^4.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Freedman's theorem gives homeomorphism to S^4. The linked census reduces its six-pentachoron S^4 cases to at most four possible PL classes and conjectures them standard. Exact unresolved remainder: Prove every smooth homotopy 4-sphere is diffeomorphic to S^4, or construct and certify an exotic one.^[1]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Freedman's theorem gives homeomorphism to S^4. The linked census reduces its six-pentachoron S^4 cases to at most four possible PL classes and conjectures them standard. Exact unresolved remainder: Prove every smooth homotopy 4-sphere is diffeomorphic to S^4, or construct and certify an exotic one.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Freedman's theorem gives homeomorphism to S^4. The linked census reduces its six-pentachoron S^4 cases to at most four possible PL classes and conjectures them standard. Exact unresolved remainder: Prove every smooth homotopy 4-sphere is diffeomorphic to S^4, or construct and certify an exotic one.^[1] Relevance to this problem For Smooth four-dimensional Poincaré conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Freedman's theorem gives homeomorphism to S^4. The linked census reduces its six-pentachoron S^4 cases to at most four possible PL classes and conjectures them standard. Exact unresolved remainder: Prove every smooth homotopy 4-sphere is diffeomorphic to S^4, or construct and certify an exotic one. Open full record →
Prove that every smooth homotopy four-sphere is diffeomorphic to S^4, or construct a smooth homotopy four-sphere and prove that it is not diffeomorphic to S^4.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every smooth homotopy four-sphere is diffeomorphic to S^4, or construct a smooth homotopy four-sphere and prove that it is not diffeomorphic to S^4.^[1] Relevance to this problem For Smooth four-dimensional Poincaré conjecture, record smooth-four-dimensional-poincare-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every smooth homotopy four-sphere is diffeomorphic to S^4, or construct a smooth homotopy four-sphere and prove that it is not diffeomorphic to S^4. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every smooth homotopy four-sphere is diffeomorphic to S^4, or construct a smooth homotopy four-sphere and prove that it is not diffeomorphic to S^4. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2026 article presents the nonexistence of exotic four-spheres as the smooth four-dimensional Poincaré conjecture. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

The topological four-dimensional Poincaré theorem is known. This record concerns smooth structures and recurring claimed proofs require specialist review.

Computational notes

Triangulation censuses can eliminate small combinatorial candidates without classifying all smooth homotopy four-spheres.

2See also

Torsion-freeness for king-grid independence complexestopology
Integral torsion in scale-four hypercube Rips complexestopology
Hodge conjecturetopology

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Plain text

“Smooth four-dimensional Poincaré conjecture.” TheoremDB. P26. Problem statement; statement text SHA-256 c4cd737641073e267a3c5af1d489119d9a43d7cf12a1230c6312866e99d829f3. https://theoremdb.org/statement/?ref=P26

BibTeX

@misc{theoremdb-problem-c4cd737641073e267a3c5af1d489119d9a43d7cf12a1230c6312866e99d829f3,
  title = {{Smooth four-dimensional Poincaré conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c4cd737641073e267a3c5af1d489119d9a43d7cf12a1230c6312866e99d829f3},
  url = {https://theoremdb.org/statement/?ref=P26}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from link.springer.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Small Triangulations of 4-Manifolds and the 4-Manifold Census, Discrete & Computational Geometry (2026). DOI 10.1007/s00454-026-00818-w. Discrete & Computational Geometry, 2026, Conjecture 1 and introduction. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited 2026 article presents the nonexistence of exotic four-spheres as the smooth four-dimensional Poincaré conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract, Conjecture 1, and census results.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the smooth four-dimensional Poincaré conjecture and supplies the finite census result without resolving it.Source named by the research packet.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P26] Smooth four-dimensional Poincaré conjecture

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Computational notes

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Background

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