[#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 \(S^4\).
1Context
Dimension four is the remaining exceptional dimension for the smooth generalized Poincaré question.
2Problem setup
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.
Remark 1. Dimension four is the remaining exceptional dimension for the smooth generalized Poincaré question.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Smooth four-dimensional Poincaré conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/smooth-four-dimensional-poincare-conjectureThis page as plain text: smooth-four-dimensional-poincare-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from link.springer.com[1], current as of July 31, 2026.
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.