# P26: Smooth four-dimensional Poincaré conjecture

- ID: `P26`
- Reference: `smooth-four-dimensional-poincare-conjecture`
- Page: https://theoremdb.org/statements/P26
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (A homotopy four-sphere).** A homotopy four-sphere is a smooth closed four-manifold with the homotopy type of S^4.
- **Definition (Diffeomorphic).** Diffeomorphic means equivalent by a smooth bijection with a smooth inverse.
- **Remark.** Dimension four is the remaining exceptional dimension for the smooth generalized Poincaré question.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited 2026 article on 2026-07-22.
- The topological four-dimensional Poincaré theorem is known. This record concerns smooth structures and recurring claimed proofs require specialist review.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `smooth-four-dimensional-poincare-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://link.springer.com/article/10.1007/s00454-026-00818-w
   - Also cited at Abstract, Conjecture 1, and census results
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.