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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Geometry.ComplexBundle.Replacement
import OAI.Geometry.ComplexBundle.SmoothLine
namespace OAI
noncomputable section
/-! Smooth bases and test bundles with replacement maps. -/
section
open Topology
open scoped Manifold ContDiff BigOperators
namespace Kaplansky
open ComplexGrassmann
abbrev SmoothTestBase (m R : ℕ) := ProjectiveProduct (testProjectiveDimension m R)
abbrev SmoothTestModel (m R : ℕ) := ProjectiveProductModel (testProjectiveDimension m R)
def testFactorProjectiveHomeomorph (m R : ℕ) (i : Fin R) :
projectionSpace (testProjectiveDimension m R i+1) 1 ≃ₜ
Grassmann 1 (testProjectiveDimension m R i) := by
let P : ℕ → Type := fun d => projectionSpace d 1 ≃ₜ Grassmann 1 (testProjectiveDimension m R i)
exact Eq.mp (congrArg P (Nat.add_comm 1 (testProjectiveDimension m R i)))
(projectiveProjectionHomeomorph (testProjectiveDimension m R i))
def testBaseProjectiveHomeomorph (m R : ℕ) : BundleTestBase m R ≃ₜ SmoothTestBase m R :=
Homeomorph.piCongrRight (testFactorProjectiveHomeomorph m R)
instance (m R : ℕ) : IsManifold 𝓘(ℝ,SmoothTestModel m R) ∞ (SmoothTestBase m R) := inferInstance
instance (m R : ℕ) : CompactSpace (SmoothTestBase m R) := inferInstance
instance (m R : ℕ) : T2Space (SmoothTestBase m R) := inferInstance
abbrev SmoothPrefixBase (m R j : ℕ) (hj : j<R) :=
ProjectiveProduct (fun i : Fin (j+1) => testProjectiveDimension m R (i.castLE hj))
abbrev SmoothPrefixModel (m R j : ℕ) (hj : j<R) :=
ProjectiveProductModel (fun i : Fin (j+1) => testProjectiveDimension m R (i.castLE hj))
theorem smoothPrefix_incidence_margin {m R j : ℕ} (hm : 0<m) (hj : j<R) :
3*Module.finrank ℝ (SmoothPrefixModel m R j hj) < 40*m*testRank m R j := by
rw [projectiveProduct_real_finrank]
have h := testPrefixDimension_margin hm hj
change 3*testPrefixDimension m R j hj < _ at h
change 3*(2*testPrefixDimension m R j hj)<_
nlinarith
end Kaplansky
end
section
open Bundle Topology
open scoped Manifold ContDiff
namespace Kaplansky
open ComplexGrassmann
abbrev stageProjectiveDimensions (m R b : ℕ) (hb : b≤R) : Fin b → ℕ :=
fun i => testProjectiveDimension m R (i.castLE hb)
/-- E_n on the first b projective factors, retaining the full-R K_i. -/
def smoothTestBundleCore (m R b n : ℕ) (hb : b≤R) (hn : n≤b) :=
smoothLineSumCore (stageProjectiveDimensions m R b hb)
(fun s : TestBundleSlot m R n => testSummandLineIndex hn s.1)
abbrev SmoothTestBundle (m R b n : ℕ) (hb : b≤R) (hn : n≤b) :=
(smoothTestBundleCore m R b n hb hn).Fiber
instance smoothTestBundle_norm (m R b n : ℕ) (hb : b≤R) (hn : n≤b)
(P : ProjectiveProduct (stageProjectiveDimensions m R b hb)) :
NormedAddCommGroup (SmoothTestBundle m R b n hb hn P) := smoothLineSum_norm _ _ P
instance smoothTestBundle_inner (m R b n : ℕ) (hb : b≤R) (hn : n≤b)
(P : ProjectiveProduct (stageProjectiveDimensions m R b hb)) :
InnerProductSpace ℂ (SmoothTestBundle m R b n hb hn P) := smoothLineSum_inner _ _ P
instance smoothTestBundle_smooth (m R b n : ℕ) (hb : b≤R) (hn : n≤b) :
ContMDiffVectorBundle ∞ (EuclideanSpace ℂ (TestBundleSlot m R n))
(SmoothTestBundle m R b n hb hn)
𝓘(ℝ,ProjectiveProductModel (stageProjectiveDimensions m R b hb)) :=
smoothLineSum_contMDiffVectorBundle _ _
theorem smoothTestBundle_finrank (m R b n : ℕ) (hb : b≤R) (hn : n≤b)
(P : ProjectiveProduct (stageProjectiveDimensions m R b hb)) :
Module.finrank ℂ (SmoothTestBundle m R b n hb hn P) = testRank m R n := by
change Module.finrank ℂ (EuclideanSpace ℂ (TestBundleSlot m R n)) = _
simpa [TestBundleSlot] using (sum_testMultiplicity (m := m) (hn.trans hb))
/-- The fresh block on Y_{j+1}, consisting of exactly a_{j+1,j+1}
occurrences of the hyperplane line of its last factor. -/
def smoothFreshCore (m R j : ℕ) (hj : j<R) :=
smoothLineSumCore (stageProjectiveDimensions m R (j+1) hj)
(fun _ : Fin (testMultiplicity m R (j+1) (Fin.last (j+1))) => some (Fin.last j))
end Kaplansky
end
section
open Bundle Topology
namespace Kaplansky
def testReplacementMultiplicity (m R j : ℕ) (i : Fin (j+1)) : ℕ :=
testMultiplicity m R (j+1) i.castSucc
def testReplacementCore (m R j : ℕ) (hj : j≤R) :=
repeatedLineCore (fun i : Fin (j+1) => testLineCore m R (testSummandLineIndex hj i))
(testReplacementMultiplicity m R j)
abbrev TestReplacementModel (m R j : ℕ) := RepeatedLineModel (testReplacementMultiplicity m R j)
theorem exists_testBundle_replacements {m R j : ℕ} (hm : 0<m) (hj : j<R) :
∃ A : ∀ y, Fin 16 → TestBundle m R j hj.le y →ₗ[ℂ]
(testReplacementCore m R j hj.le).Fiber y,
(∀ i, Continuous (fun z : TotalSpace (EuclideanSpace ℂ (TestBundleSlot m R j))
(TestBundle m R j hj.le) =>
TotalSpace.mk' (TestReplacementModel m R j) z.proj (A z.proj i z.2))) ∧
∀ y, ExpandingThrough m (A y) := by
let C := fun i : Fin (j+1) => testLineCore m R (testSummandLineIndex hj.le i)
let r := testReplacementMultiplicity m R j
have hr : ∀ i,16*m≤r i := by
intro i
have hq := testMultiplicity_quarter (m := m) hj i
have hl := testMultiplicity_lower_bound (m := m) hj i
change 16*m≤testMultiplicity m R (j+1) i.castSucc
omega
have he : (fun i : Fin (j+1) => 4*r i) = testMultiplicity m R j := by
funext i
exact testMultiplicity_quarter hj i
let P : (Fin (j+1) → ℕ) → Prop := fun D =>
∃ A : ∀ y, Fin 16 → (repeatedLineCore C D).Fiber y →ₗ[ℂ] (repeatedLineCore C r).Fiber y,
(∀ i, Continuous (fun z : TotalSpace (RepeatedLineModel D) (repeatedLineCore C D).Fiber =>
TotalSpace.mk' (RepeatedLineModel r) z.proj (A z.proj i z.2))) ∧
∀ y, ExpandingThrough m (A y)
have h : P (fun i => 4*r i) := exists_continuous_repeatedLine_replacements C m hm r hr
exact Eq.mp (congrArg P he) h
end Kaplansky
end
open Bundle Topology
namespace Kaplansky
open ComplexGrassmann
def smoothReplacementCore (m R j : ℕ) (hj : j<R) :=
repeatedLineCore (fun i : Fin (j+1) =>
productLineCore (stageProjectiveDimensions m R (j+1) hj)
(testSummandLineIndex (Nat.le_succ j) i)) (testReplacementMultiplicity m R j)
theorem exists_smoothTest_replacements {m R j : ℕ} (hm : 0<m) (hj : j<R) :
∃ A : ∀ y, Fin 16 → SmoothTestBundle m R (j+1) j hj (Nat.le_succ j) y →ₗ[ℂ]
(smoothReplacementCore 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' (TestReplacementModel m R j) z.proj (A z.proj i z.2))) ∧
∀ y, ExpandingThrough m (A y) := by
let C := fun i : Fin (j+1) => productLineCore (stageProjectiveDimensions m R (j+1) hj)
(testSummandLineIndex (Nat.le_succ j) i)
let r := testReplacementMultiplicity m R j
have hr : ∀ i,16*m≤r i := by
intro i
have hq := testMultiplicity_quarter (m := m) hj i
have hl := testMultiplicity_lower_bound (m := m) hj i
change 16*m≤testMultiplicity m R (j+1) i.castSucc
omega
have he : (fun i : Fin (j+1) => 4*r i) = testMultiplicity m R j := by
funext i
exact testMultiplicity_quarter hj i
let P : (Fin (j+1) → ℕ) → Prop := fun D =>
∃ A : ∀ y, Fin 16 → (repeatedLineCore C D).Fiber y →ₗ[ℂ] (repeatedLineCore C r).Fiber y,
(∀ i, Continuous (fun z : TotalSpace (RepeatedLineModel D) (repeatedLineCore C D).Fiber =>
TotalSpace.mk' (RepeatedLineModel r) z.proj (A z.proj i z.2))) ∧
∀ y, ExpandingThrough m (A y)
have h : P (fun i => 4*r i) := exists_continuous_repeatedLine_replacements C m hm r hr
exact Eq.mp (congrArg P he) h
end Kaplansky
end
end OAI
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