feat: add generalized topological Poincaré in dim ≥ 5 (Smale) eval problem

LeanEval/Topology/PoincareHighDimTopological.lean

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+import Mathlib
+import EvalTools.Markers
+
+namespace LeanEval
+namespace Topology
+
+/-!
+# Generalized topological Poincaré conjecture in dimensions ≥ 5 (Smale)
+
+A specialization of mathlib's generalized `proof_wanted
+ContinuousMap.HomotopyEquiv.nonempty_homeomorph_sphere` in
+`Mathlib/Geometry/Manifold/PoincareConjecture.lean`, restricted to
+dimension `n ≥ 5`: every Hausdorff `n`-manifold homotopy-equivalent to the
+standard `n`-sphere `𝕊ⁿ` is homeomorphic to `𝕊ⁿ`.
+
+This is **Stephen Smale's 1961 theorem**, originally proved for `n ≥ 5`
+assuming a smooth structure on `M` (via the `h`-cobordism theorem). The
+**topological** version (no smoothness assumption) was completed by
+Newman in 1966 and Connell in 1967. Smale received the Fields Medal in
+1966.
+
+mathlib has homotopy equivalences (`≃ₕ`), homeomorphisms (`≃ₜ`),
+`ChartedSpace`, and the unit sphere as a topological space — but no
+`h`-cobordism theorem, no handle decompositions, and no Poincaré
+conjecture in any dimension.
+-/
+
+open Metric (sphere)
+open ContinuousMap
+
+/-- **Generalized topological Poincaré conjecture in dimensions ≥ 5**
+(Smale 1961; topological case Newman 1966 / Connell 1967). For `n ≥ 5`,
+every Hausdorff `n`-manifold homotopy-equivalent to the standard `n`-sphere
+`𝕊ⁿ` is homeomorphic to `𝕊ⁿ`. -/
+@[eval_problem]
+theorem poincare_high_dim_topological
+    {n : ℕ} (_h5 : 5 ≤ n)
+    {M : Type*} [TopologicalSpace M] [T2Space M]
+    [ChartedSpace (EuclideanSpace ℝ (Fin n)) M]
+    (_h : M ≃ₕ sphere (0 : EuclideanSpace ℝ (Fin (n + 1))) 1) :
+    Nonempty (M ≃ₜ sphere (0 : EuclideanSpace ℝ (Fin (n + 1))) 1) := by
+  sorry
+
+end Topology
+end LeanEval

manifests/problems/poincare_high_dim_topological.toml

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+id = "poincare_high_dim_topological"
+title = "Generalized topological Poincaré conjecture in dimensions ≥ 5 (Smale)"
+test = false
+module = "LeanEval.Topology.PoincareHighDimTopological"
+holes = ["poincare_high_dim_topological"]
+submitter = "Kim Morrison"
+notes = "For n ≥ 5, every Hausdorff n-manifold homotopy-equivalent to 𝕊ⁿ is homeomorphic to 𝕊ⁿ. Specialization to n ≥ 5 of mathlib's generalized `proof_wanted ContinuousMap.HomotopyEquiv.nonempty_homeomorph_sphere` in `Mathlib/Geometry/Manifold/PoincareConjecture.lean`. Stephen Smale 1961 (Bull. AMS 66, Ann. of Math. 74) originally for smooth M via the h-cobordism theorem; topological version completed by Newman 1966 and Connell 1967. Smale received the Fields Medal in 1966. Mathlib has the homotopy-equiv / homeo / ChartedSpace scaffolding but no h-cobordism theorem, no handle decompositions, and no Poincaré conjecture in any dimension."
+source = "S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391-406. Topological case: M. H. A. Newman, The engulfing theorem for topological manifolds, Ann. of Math. 84 (1966) and E. H. Connell, A topological h-cobordism theorem for n ≥ 5, Illinois J. Math. 11 (1967). Specialization of `proof_wanted ContinuousMap.HomotopyEquiv.nonempty_homeomorph_sphere` in Mathlib/Geometry/Manifold/PoincareConjecture.lean."
+informal_solution = "Smale's smooth proof: a smooth homotopy n-sphere M (n ≥ 5) bounds a smooth contractible (n+1)-manifold W via Whitehead's theorem and a Mayer-Vietoris computation; apply the h-cobordism theorem (Smale's main contribution) to W minus two small balls to conclude W ≃ D^{n+1}, hence M = ∂W ≃ S^n smoothly. The topological version (Newman/Connell) replaces the h-cobordism step with the topological engulfing theorem and a topological h-cobordism theorem, proving the same conclusion without the smoothness hypothesis. The dim ≥ 5 hypothesis is crucial: it allows the Whitney trick to remove pairs of intersections, which fails in dim 4 (requiring Casson handles, Freedman 1982) and below."