The simply connected endgame for a standard connected-sum decomposition
ProvedPoincareFormalization.ExtinctionEndgame.nonempty_homeomorph_sphere_of_standard_decompositionconnected-sumfinite-extinctionopenga-endgamepoincare-conjecture
A closed simply connected three-manifold with a finite connected-sum decomposition into sphere-covered factors and S1 times S2 factors is homeomorphic to S3. This conditional proof sketch uses three open topology lemmas: simple connectivity descends to factors, sphere handles are not simply connected, and the sum of two three-spheres is a three-sphere. The sphere-covering base case reuses salim's proved covering theorem.
Preamble
import Definitions.Def_OpenGA_SurgeryTopologyEvolution
Formal statement
/-!
# The formal finite-extinction reduction of the Poincare goal
The proofs in this file have no holes of their own, but depend on the four
explicit open theorems in `OpenProblems.lean`. They are conditional proof
sketches, not proofs of the Poincare conjecture or of geometric extinction.
Dependency chain:
geometric construction (open) -> width-controlled component evolution
-> finite extinction -> connected-sum reconstruction
-> simply connected endgame (three open topology lemmas) -> mission goal.
The auxiliary definitions and analytic/topological iteration theorems in
OpenGALib have no project axioms or proof holes. The distinction is checked
by `Audit.lean` and the reusable-library regression check.
-/
namespace PoincareFormalization.ExtinctionEndgame
open OpenGA
universe u
theorem nonempty_homeomorph_sphere_of_standard_decomposition
(M : ClosedThreeManifold.{u}) [SimplyConnectedSpace M]
(h : ConnectedSumClosure ClosedThreeManifold.IsStandardFactor M) :
Nonempty (M ≃ₜ SphereThree) := by sorry
end PoincareFormalization.ExtinctionEndgame
Source