Poincaré conjecture in dimension 2

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SmoothPoincare4.poincare_dim_two

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

4-manifoldspoincare-conjecturetopology

Let $M$ be a compact Hausdorff smooth $2$ -manifold without boundary. If $M$ is homotopy equivalent to $S^2$ , then $M$ is diffeomorphic to $S^2$ .

Preamble

import Mathlib

open scoped Manifold ContDiff
open Metric

Formal statement

namespace SmoothPoincare4

theorem poincare_dim_two (M : Type) [TopologicalSpace M] [T2Space M] [CompactSpace M]
    [ChartedSpace (EuclideanSpace ℝ (Fin 2)) M] [IsManifold (𝓡 2) ∞ M]
    (h : Nonempty (ContinuousMap.HomotopyEquiv M (sphere (0 : EuclideanSpace ℝ (Fin 3)) 1))) :
    Nonempty (M ≃ₘ^∞⟮𝓡 2, 𝓡 2⟯ sphere (0 : EuclideanSpace ℝ (Fin 3)) 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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