diff --git a/LeanEval/Topology/Poincare3DSmooth.lean b/LeanEval/Topology/Poincare3DSmooth.lean new file mode 100644 index 0000000..13543ac --- /dev/null +++ b/LeanEval/Topology/Poincare3DSmooth.lean @@ -0,0 +1,42 @@ +import Mathlib +import EvalTools.Markers + +namespace LeanEval +namespace Topology + +/-! +# 3D smooth Poincaré conjecture (Perelman) + +A `proof_wanted` in `Mathlib/Geometry/Manifold/PoincareConjecture.lean`. Every +simply connected compact Hausdorff 3-manifold that admits a smooth structure +is **diffeomorphic** to the standard 3-sphere `𝕊³`. + +This is the smooth-category strengthening of the topological 3D Poincaré +conjecture, also proved by Perelman in 2002–2003. In dimension 3 the smooth +and topological Poincaré conjectures are equivalent (every topological +3-manifold admits a unique smooth structure, by Moise's theorem), but the +formal Lean statement carries the smooth structure explicitly via +`IsManifold (𝓡 3) ∞ M` and concludes with a diffeomorphism in the manifold +type `M ≃ₘ⟮𝓡 3, 𝓡 3⟯ 𝕊³`. + +mathlib has `ChartedSpace`, `IsManifold`, `Diffeomorph`, `SimplyConnectedSpace`, +`CompactSpace`, and the unit sphere as a smooth manifold but no Ricci flow, +no Hamilton–Perelman surgery, and no Moise theorem. +-/ + +open scoped Manifold ContDiff +open Metric (sphere) + +/-- **3D smooth Poincaré conjecture** (Perelman, 2002–2003). Every simply +connected compact Hausdorff smooth 3-manifold is diffeomorphic to the +standard 3-sphere `𝕊³`. -/ +@[eval_problem] +theorem poincare_3d_smooth + {M : Type*} [TopologicalSpace M] [T2Space M] + [ChartedSpace (EuclideanSpace ℝ (Fin 3)) M] [IsManifold (𝓡 3) ∞ M] + [SimplyConnectedSpace M] [CompactSpace M] : + Nonempty (M ≃ₘ⟮𝓡 3, 𝓡 3⟯ sphere (0 : EuclideanSpace ℝ (Fin 4)) 1) := by + sorry + +end Topology +end LeanEval diff --git a/manifests/problems/poincare_3d_smooth.toml b/manifests/problems/poincare_3d_smooth.toml new file mode 100644 index 0000000..4e720f5 --- /dev/null +++ b/manifests/problems/poincare_3d_smooth.toml @@ -0,0 +1,9 @@ +id = "poincare_3d_smooth" +title = "3D smooth Poincaré conjecture (Perelman)" +test = false +module = "LeanEval.Topology.Poincare3DSmooth" +holes = ["poincare_3d_smooth"] +submitter = "Kim Morrison" +notes = "Every simply connected compact Hausdorff smooth 3-manifold is diffeomorphic to 𝕊³. Recorded as `proof_wanted SimplyConnectedSpace.nonempty_diffeomorph_sphere_three` in `Mathlib/Geometry/Manifold/PoincareConjecture.lean`. The smooth-category strengthening of the topological 3D Poincaré conjecture; also Perelman 2002-2003. In dim 3 the smooth and topological versions are equivalent (Moise's theorem says every topological 3-manifold admits a unique smooth structure), but the Lean statement carries the smooth structure explicitly. Mathlib has all the differential-topology / manifold scaffolding but neither Ricci flow nor Moise's theorem." +source = "G. Perelman, three arXiv preprints 2002-2003 (math/0211159, math/0303109, math/0307245). Recorded as a `proof_wanted` in Mathlib/Geometry/Manifold/PoincareConjecture.lean. In dim 3, Smooth ⇔ PL ⇔ Topological via Moise (1952) / Bing (1959)." +informal_solution = "Apply the topological 3D Poincaré conjecture to obtain a homeomorphism M ≃ₜ S³, then promote it to a diffeomorphism via Moise's theorem (every topological 3-manifold admits a unique smooth structure up to diffeomorphism), or equivalently observe that Hamilton-Perelman Ricci flow with surgery runs in the smooth category and produces a smooth diffeomorphism directly. Either route involves the same Perelman input on the geometric side; the dim-3 special feature is Moise/Bing which collapses the smooth/topological distinction (false in dim 4 and above)."