| import OAI.Analysis.Kaplansky.PolynomialTest |
| import OAI.Geometry.ComplexBundle.Complement |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| /-! Regrouping retained frames and their complement bundles. -/ |
|
|
| section |
| open Bundle |
| open scoped BigOperators |
| namespace Kaplansky |
| open ComplexGrassmann |
| attribute [local instance 2000] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing |
|
|
| namespace ContinuousOrthonormalStageFrame |
| variable {ι : Type*} [Fintype ι] {K : ι → ℕ} {a b : LineStage K} {d : ℕ} |
|
|
| def congr (f : ContinuousOrthonormalStageFrame a d) |
| (e : UnitaryBundleEquiv a.Model b.Model a.Fiber b.Fiber) : |
| ContinuousOrthonormalStageFrame b d where |
| vector y i := e.fiber y (f.vector y i) |
| continuous_vector i := e.continuous_forward.comp (f.continuous_vector i) |
| orthonormal y := ((e.fiber y).toLinearIsometry.orthonormal_comp_iff).mpr (f.orthonormal y) |
| end ContinuousOrthonormalStageFrame |
|
|
| def retainedGroupedStage (m R : ℕ) : LineStage (testProjectiveDimension m R) := |
| ⟨RetainedGroupedSlot m R,inferInstance,fun s => s.1⟩ |
|
|
| / |
| presentation used in the Chern computation. No characteristic-class |
| nonvanishing or obstruction is a premise. -/ |
| theorem properlyInfinite_regrouped_frame {m : ℕ} (hm : 0<m) |
| (hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) : |
| ∃ R : ℕ, 0<R ∧ Nonempty (ContinuousOrthonormalStageFrame (retainedGroupedStage m R) |
| (2*m*testRank m R 0)) := by |
| obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_retained_frame hm hpi |
| exact ⟨R,hR,⟨f.congr (retainedRegroupingUnitary m R)⟩⟩ |
|
|
| / |
| frame whose orthogonal complement has rank below the Chern degree on every |
| fiber. Identifying that Chern class and proving it nonzero remain separate. -/ |
| theorem properlyInfinite_retained_complement {m : ℕ} (hm : 0<m) |
| (hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) : |
| ∃ (R : ℕ) (_ : 0<R) (f : ContinuousOrthonormalStageFrame (retainedLineStage m R) |
| (2*m*testRank m R 0)), |
| ∀ y : SmoothTestBase m R, |
| Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ < |
| ∑ i : Fin R, testProjectiveDimension m R i := by |
| obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_retained_frame hm hpi |
| refine ⟨R,hR,f,fun y => ?_⟩ |
| exact retained_complement_rank_lt hm y _ (f.image_finrank y).ge |
|
|
| end Kaplansky |
|
|
| end |
|
|
| open Bundle Topology |
| open scoped BigOperators |
| namespace Kaplansky |
| open ComplexGrassmann |
| attribute [local instance 2000] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing |
|
|
| theorem retainedGrouped_complementBundle_rank_lt {m R : ℕ} (hm : 0<m) |
| (f : ContinuousOrthonormalStageFrame (retainedGroupedStage m R) (2*m*testRank m R 0)) |
| (y : SmoothTestBase m R) : |
| Module.finrank ℂ (f.ComplementFiber y) < ∑ i : Fin R, testProjectiveDimension m R i := by |
| have hr := f.complement_finrank y |
| have hc : Fintype.card (retainedGroupedStage m R).Slot = retainedTrivialMultiplicity m R + |
| ∑ i : Fin R, testProjectiveDimension m R i := by |
| have hd := retainedBundle_rank_decomposition m R y |
| change Module.finrank ℂ (EuclideanSpace ℂ (RetainedSlot m R)) = _ at hd |
| rw [finrank_euclideanSpace, Fintype.card_congr (retainedGroupingEquiv m R)] at hd |
| exact hd |
| rw [hc] at hr |
| have ht := retainedTrivialMultiplicity_bound (R := R) hm |
| change Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ < _ |
| omega |
|
|
| / |
| the ordinary complement vector bundle and the actual global bundle splitting. |
| No characteristic-class hypothesis, bundle realization or norm bound is input. -/ |
| theorem properlyInfinite_retained_bundle_split {m : ℕ} (hm : 0<m) |
| (hpi : ProperlyInfinite (CStarMatrix (Fin m) (Fin m) ActualAlgebra)) : |
| ∃ (R : ℕ) (_ : 0<R) |
| (f : ContinuousOrthonormalStageFrame (retainedGroupedStage m R) (2*m*testRank m R 0)) |
| (H : f.ComplementSplitTotal ≃ₜ |
| TotalSpace (retainedGroupedStage m R).Model (retainedGroupedStage m R).Fiber), |
| (∀ z, (H z).proj=z.1.proj) ∧ |
| (∀ y : SmoothTestBase m R, Module.finrank ℂ (f.ComplementFiber y) < |
| ∑ i : Fin R, testProjectiveDimension m R i) := by |
| obtain ⟨R,hR,⟨f⟩⟩ := properlyInfinite_regrouped_frame hm hpi |
| exact ⟨R,hR,f,f.complementSplitHomeomorph,f.complementSplitHomeomorph_base, |
| retainedGrouped_complementBundle_rank_lt hm f⟩ |
|
|
| end Kaplansky |
|
|
| end |
|
|
| end OAI |
|
|
