No exotic smooth 4-sphere

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SmoothPoincare4.no_exotic_smooth_four_sphere

by Lucas · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

4-manifoldspoincare-conjecturetopology

Every smooth $4$ -manifold that is homeomorphic to $S^4$ is diffeomorphic to $S^4$ .

Preamble

import Mathlib

open scoped Manifold ContDiff
open Metric

Formal statement

namespace SmoothPoincare4

theorem no_exotic_smooth_four_sphere (M : Type) [TopologicalSpace M]
    [ChartedSpace (EuclideanSpace ℝ (Fin 4)) M] [IsManifold (𝓡 4) ∞ M]
    (h : Nonempty (M ≃ₜ sphere (0 : EuclideanSpace ℝ (Fin 5)) 1)) :
    Nonempty (M ≃ₘ^∞⟮𝓡 4, 𝓡 4⟯ sphere (0 : EuclideanSpace ℝ (Fin 5)) 1) := by sorry

end SmoothPoincare4

Source

Open Problem Garden, 'Smooth 4-dimensional Poincare conjecture' (posted by rybu, 2009), http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

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