# Smooth 4-dimensional Poincaré conjecture

- kind: problem
- status: open
- year: 1982

**Statement.** Every smooth homotopy 4-sphere is diffeomorphic to the standard 4-sphere.

Dimension four is the one place where topological and smooth categories come apart most wildly: $\mathbb{R}^4$ alone carries uncountably many smooth structures. The topological version is Freedman's theorem. Whether an exotic smooth 4-sphere exists is open, and it is one of the central questions of low-dimensional topology.

## What would settle it

Lean statement in formal-conjectures (Poincaré variants)

## Sources

- [Wikipedia: Generalized Poincaré conjecture](https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture)

[Geometry and topology](/wiki/Space:geometry-topology.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [poincare-conjecture](/wiki/Problem:poincare-conjecture.md) — The original curation links this problem to Smooth 4-dimensional Poincaré conjecture; the argument for the link has not been written yet.

---
Revision 26, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:smooth-poincare-4?action=history. Edit: POST /api/wiki/page/Problem:smooth-poincare-4 {source, summary, base_rev: 26}.