| import OAI.Dynamics.EntropyCounterexample.MapFinalReduction |
| import OAI.Geometry.SingularHomology.SphereSquareHomologyTransport |
| import OAI.Geometry.SingularHomology.FiniteCoverHomologyRawBase |
| import OAI.Geometry.SingularHomology.SphereMayerVietorisEigen |
| import OAI.Dynamics.EntropyCounterexample.SmallCoverNaturality |
| import OAI.Geometry.SingularHomology.SingularSmallComparison |
| import OAI.Geometry.SingularHomology.SingularSubdivisionEventuallySmall |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| open Problem340 Problem340.SphereCover Problem340.SingularSmallChains |
| open Problem340.SmallCoverMayerVietoris Problem340.SphereMayerVietorisEigen |
| namespace Problem340.CounterexampleAssembly804 |
|
|
| theorem sphereEigen : ∃ v : RealSingularHomology 2 StandardSphere2, v ≠ 0 ∧ |
| realHomologyMap 2 SphereDoubling.sphereMap SphereDoubling.continuous_sphereMap v = |
| (2 : ℝ) • v := by |
| let : QuasiIso (inclusion coverFamily) := |
| inclusion_quasiIso_of_natural_subdivision_zero coverFamily |
| ActualSingularSubdivision.chainMap ActualSingularSubdivision.chainMap_app_zero |
| (exists_actualSubdivision_pow_mem_range coverFamily coverFamily_isOpen |
| (by rw [iUnion_coverFamily])) |
| let e := (singularConnectingIsoOfQuasiIsoAt northComplement southComplement 1 |
| (by decide)).toLinearEquiv |
| obtain ⟨w, hw, heigen⟩ := exists_overlap_square_eigenvector |
| refine ⟨e.symm w, (map_ne_zero_iff e.symm e.symm.injective).2 hw, ?_⟩ |
| apply e.injective |
| have h := singularConnectingIsoOfQuasiIsoAt_naturality_apply |
| SphereDoubling.sphereMap SphereDoubling.continuous_sphereMap |
| northComplement southComplement northComplement southComplement |
| mapsTo_northComplement mapsTo_southComplement 1 (by decide) (e.symm w) |
| have hh : e (realHomologyMap 2 SphereDoubling.sphereMap |
| SphereDoubling.continuous_sphereMap (e.symm w)) = |
| realHomologyMap 1 SphereOverlapDoubling.overlapSquare |
| SphereOverlapDoubling.continuous_overlapSquare (e (e.symm w)) := by |
| simpa only [e, CategoryTheory.Iso.toLinearEquiv_apply, intersectionMap, restrictMap_eq_overlapSquare] using h |
| rw [hh, e.apply_symm_apply, heigen, map_smul, e.apply_symm_apply] |
|
|
| theorem exists_counterexample : |
| ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, ∃ hf : IsC1SelfMap q f, Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ (∃ v : RealSingularHomology 2 (CounterexampleManifold q), v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ 0 < Real.log (2 : ℝ) ∧ Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := by |
| have hsphere : ∃ v : RealSingularHomology 2 (OnePoint ℂ), v ≠ 0 ∧ |
| realHomologyMap 2 rawSphereSquare continuous_rawSphereSquare v = (2 : ℝ) • v := by |
| obtain ⟨v, hv, he⟩ := sphereEigen |
| exact rawSphereSquare_eigenvector_of_standard 2 2 v hv he |
| let G := CoordinateGoodCover.rawBaseGoodCover |
| let : QuasiIso (inclusion G.carrier) := |
| inclusion_quasiIso_of_natural_subdivision_zero G.carrier |
| ActualSingularSubdivision.chainMap ActualSingularSubdivision.chainMap_app_zero |
| (exists_actualSubdivision_pow_mem_range G.carrier G.isOpen |
| (by rw [G.iUnion_eq_univ])) |
| exact Construction.exists_counterexample_of_sphere_and_base_homology hsphere |
| FiniteCoverHomology.rawBase_homologyEigenvalueRadii_bddAbove |
|
|
| end Problem340.CounterexampleAssembly804 |
|
|
| end |
|
|
| end OAI |
|
|
