Conjecture Fieldbook

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Smooth 4D Poincaré conjecture

Canonical problem dossier

Is every smooth four-manifold homeomorphic to the four-sphere also diffeomorphic to the standard four-sphere?

Status
Open
Field
Topology
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

M4≃S4⟹?M4≅diffS4M^4\simeq S^4\stackrel{?}{\Longrightarrow}M^4\cong_{\mathrm{diff}}S^4

Why it matters

Dimension four has exceptional smooth behavior. The answer would clarify whether an exotic smooth structure exists on the 4-sphere.

Context and known approaches

The topological four-dimensional Poincaré conjecture is known, but the corresponding smooth statement remains unresolved.

Formal boundary

If a smooth closed 4-manifold is homotopy equivalent to S⁴, must it be diffeomorphic to S⁴?

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Wikipedia — Dimension-by-dimension status.
  2. Wikipedia open-problems list — Notes the smooth four-dimensional generalization remains open.