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8e4d533 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import OAI.Geometry.BundleTests.Smooth
import OAI.Geometry.ComplexBundle.CoordinatesMetric
namespace OAI
noncomputable section
/-! Fresh bundle maps and assembly of the test-bundle step. -/
section
open Bundle Topology Module
open scoped Manifold ContDiff
namespace Kaplansky
open ComplexGrassmann
theorem testBundleSlot_card {m R j : ℕ} (hj : j≤R) :
Fintype.card (TestBundleSlot m R j) = testRank m R j := by
simpa [TestBundleSlot] using (sum_testMultiplicity (m := m) hj)
abbrev TestSourceCoordinates (m R j : ℕ) (hj : j<R) :=
FiniteCoordinateBundle (stageProjectiveDimensions m R (j+1) hj)
(fun s : TestBundleSlot m R j => testSummandLineIndex (Nat.le_succ j) s.1)
(testRank m R j) (testBundleSlot_card (Nat.le_of_lt hj))
abbrev TestFreshRank (m R j : ℕ) := testMultiplicity m R (j+1) (Fin.last (j+1))
abbrev TestFreshCoordinates (m R j : ℕ) (hj : j<R) :=
FiniteCoordinateBundle (stageProjectiveDimensions m R (j+1) hj)
(fun _ : Fin (TestFreshRank m R j) => some (Fin.last j))
(TestFreshRank m R j) (Fintype.card_fin _)
theorem exists_test_eight_fresh_bundle_maps {m R j : ℕ} (hm : 0<m) (hj : j<R) :
∃ T : Fin 8 → ∀ y, TestSourceCoordinates m R j hj y →L[ℂ] TestFreshCoordinates m R j hj y,
(∀ i, ContMDiff 𝓘(ℝ,SmoothPrefixModel m R j hj)
(𝓘(ℝ,SmoothPrefixModel m R j hj).prod
𝓘(ℝ,(Fin (testRank m R j) → ℂ) →L[ℝ] (Fin (TestFreshRank m R j) → ℂ))) ∞
(fun y => TotalSpace.mk' ((Fin (testRank m R j) → ℂ) →L[ℝ]
(Fin (TestFreshRank m R j) → ℂ)) y ((T i y).restrictScalars ℝ))) ∧
∀ y (S : Submodule ℂ (TestSourceCoordinates m R j hj y)), m<finrank ℂ S →
2*finrank ℂ S≤finrank ℂ (⨆ i, S.map (T i y).toLinearMap :
Submodule ℂ (TestFreshCoordinates m R j hj y)) := by
apply Incidence.exists_eight_fresh_bundle_maps
· unfold testRank
positivity
· exact testMultiplicity_fresh hj
· exact smoothPrefix_incidence_margin hm hj
end Kaplansky
end
section
open Bundle Topology Module
namespace Kaplansky
open ComplexGrassmann
/-- The finite-coordinate incidence maps give genuine continuous maps on
the original Hilbert-sum presentation of E_j and its fresh block. -/
theorem exists_smoothTest_freshMaps {m R j : ℕ} (hm : 0<m) (hj : j<R) :
∃ A : ∀ y, Fin 8 → SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y →ₗ[ℂ]
(smoothFreshCore m R j hj).Fiber y,
(∀ i, Continuous (fun z : TotalSpace (EuclideanSpace ℂ (TestBundleSlot m R j))
(SmoothTestBundle m R (j+1) j hj (Nat.le_succ j)) =>
TotalSpace.mk' (EuclideanSpace ℂ (Fin (TestFreshRank m R j))) z.proj (A z.proj i z.2))) ∧
∀ y (S : Submodule ℂ (SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y)),
m<finrank ℂ S → 2*finrank ℂ S≤finrank ℂ
(⨆ i, S.map (A y i) : Submodule ℂ ((smoothFreshCore m R j hj).Fiber y)) := by
obtain ⟨T,hT,hExp⟩ := exists_test_eight_fresh_bundle_maps hm hj
let e := finiteCoordinateEquiv (testRank m R j) (testBundleSlot_card (Nat.le_of_lt hj) (m := m))
let f := finiteCoordinateEquiv (TestFreshRank m R j) (Fintype.card_fin (TestFreshRank m R j))
let a := fun y => (show SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y ≃ₗ[ℂ]
TestSourceCoordinates m R j hj y from e.toLinearEquiv)
let b := fun y => (show (smoothFreshCore m R j hj).Fiber y ≃ₗ[ℂ]
TestFreshCoordinates m R j hj y from f.toLinearEquiv)
let A := fun y i => (b y).symm.toLinearMap.comp ((T i y).toLinearMap.comp (a y).toLinearMap)
refine ⟨A,?_,?_⟩
· intro i
have hc : Continuous (fun z : TotalSpace (Fin (testRank m R j) → ℂ) (TestSourceCoordinates m R j hj) =>
TotalSpace.mk' (Fin (TestFreshRank m R j) → ℂ) z.proj (T i z.proj z.2)) :=
((hT i).continuous.comp (FiberBundle.continuous_proj _ _)).clm_bundle_apply continuous_id
let H := finiteCoordinateTotalHomeomorph (stageProjectiveDimensions m R (j+1) hj)
(fun s : TestBundleSlot m R j => testSummandLineIndex (Nat.le_succ j) s.1)
(testRank m R j) (testBundleSlot_card (Nat.le_of_lt hj))
let G := finiteCoordinateTotalHomeomorph (stageProjectiveDimensions m R (j+1) hj)
(fun _ : Fin (TestFreshRank m R j) => some (Fin.last j))
(TestFreshRank m R j) (Fintype.card_fin _)
exact G.symm.continuous.comp (hc.comp H.continuous)
· intro y S hS
have h := hExp y (S.map (a y).toLinearMap) (by simpa only [(a y).finrank_map_eq] using hS)
rw [(a y).finrank_map_eq] at h
have he : (⨆ i, S.map (A y i)) =
(⨆ i, (S.map (a y).toLinearMap).map (T i y).toLinearMap).map (b y).symm.toLinearMap := by
rw [Submodule.map_iSup]
apply iSup_congr
intro i
rw [Submodule.map_comp,Submodule.map_comp]
rw [he,(b y).symm.finrank_map_eq]
exact h
end Kaplansky
end
open Bundle Topology
namespace Kaplansky
open ComplexGrassmann
abbrev TestNextSlot (m R j : ℕ) :=
(Σ i : Fin (j+1), Fin (testReplacementMultiplicity m R j i)) ⊕ Fin (TestFreshRank m R j)
def testNextLabel (m R j : ℕ) : TestNextSlot m R j → Option (Fin (j+1)) :=
Sum.elim (fun s => testSummandLineIndex (Nat.le_succ j) s.1) (fun _ => some (Fin.last j))
/-- E_{j+1} presented as the direct sum of its replacement and fresh blocks. -/
def smoothNextCore (m R j : ℕ) (hj : j<R) :=
smoothLineSumCore (stageProjectiveDimensions m R (j+1) hj) (testNextLabel m R j)
theorem testNextSlot_card {m R j : ℕ} (hj : j<R) :
Fintype.card (TestNextSlot m R j) = testRank m R (j+1) := by
have h := sum_testMultiplicity (m := m) hj
rw [Fin.sum_univ_castSucc] at h
change Fintype.card ((Σ i : Fin (j+1), Fin (testReplacementMultiplicity m R j i)) ⊕
Fin (TestFreshRank m R j)) = _
rw [Fintype.card_sum, Fintype.card_sigma]
calc
_ = (∑ i : Fin (j+1), testReplacementMultiplicity m R j i) + TestFreshRank m R j := by
apply congrArg₂ (fun a b : ℕ => a + b)
· exact Finset.sum_congr rfl (fun i _ => Fintype.card_fin _)
· exact Fintype.card_fin _
_ = _ := h
theorem exists_test_twentyFour_expanding_maps {m R j : ℕ} (hm : 0<m) (hj : j<R) :
∃ T : ∀ y, Fin 24 → SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y →ₗ[ℂ]
(smoothNextCore m R j hj).Fiber y,
(∀ i, Continuous (fun z : TotalSpace (EuclideanSpace ℂ (TestBundleSlot m R j))
(SmoothTestBundle m R (j+1) j hj (Nat.le_succ j)) =>
TotalSpace.mk' (EuclideanSpace ℂ (TestNextSlot m R j)) z.proj (T z.proj i z.2))) ∧
∀ y, IntrinsicExpanding (T y) := by
obtain ⟨A,hA,heA⟩ := exists_smoothTest_replacements hm hj
obtain ⟨B,hB,heB⟩ := exists_smoothTest_freshMaps hm hj
let L := fun u : TestNextSlot m R j =>
productLineCore (stageProjectiveDimensions m R (j+1) hj) (testNextLabel m R j u)
let T : ∀ y, Fin 24 → SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y →ₗ[ℂ]
(smoothNextCore m R j hj).Fiber y := fun y q => Fin.addCases
(motive := fun _ : Fin (16+8) => SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y →ₗ[ℂ] (smoothNextCore m R j hj).Fiber y)
(fun i => (lineSumInl L y).comp (A y i))
(fun i => (lineSumInr L y).comp (B y i)) q
refine ⟨T,?_,?_⟩
· intro i
refine Fin.addCases (m := 16) (n := 8) ?_ ?_ i
· intro a
have ht (y) : T y (Fin.castAdd 8 a) = (lineSumInl L y).comp (A y a) :=
Fin.addCases_left a
apply ((continuous_lineSumInl L).comp (hA a)).congr
intro z
apply congrArg (TotalSpace.mk' _ z.proj)
exact (congrArg (fun F => F z.2) (ht z.proj)).symm
· intro b
have ht (y) : T y (Fin.natAdd 16 b) = (lineSumInr L y).comp (B y b) :=
Fin.addCases_right b
apply ((continuous_lineSumInr L).comp (hB b)).congr
intro z
apply congrArg (TotalSpace.mk' _ z.proj)
exact (congrArg (fun F => F z.2) (ht z.proj)).symm
· intro y
let : FiniteDimensional ℂ ((finiteLineSumCore L).Fiber y) :=
inferInstanceAs (FiniteDimensional ℂ (EuclideanSpace ℂ (TestNextSlot m R j)))
exact combined_expanding m (A y) (B y) (lineSumInl L y) (lineSumInr L y)
(lineSumInl_injective L y) (lineSumInr_injective L y) (heA y) (heB y)
end Kaplansky
end
end OAI
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