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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
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