| import Mathlib |
| import OAI.Geometry.Borsuk.Support.BalancedChart |
| import OAI.Geometry.Borsuk.Support.MixedHits |
| import OAI.Geometry.Borsuk.Projectors.CompactTargetHits |
| import OAI.Geometry.Borsuk.MatrixGeometry.TraceHomotopy |
| import OAI.Geometry.Borsuk.Projectors.CompactLocalHomotopy |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| open Set |
| section ChartSurgery |
| variable {ν ι κ : Type*} [Fintype ν] [LinearOrder ν] [Nonempty ν] [Fintype ι] [DecidableEq ι] [Fintype κ] |
| local notation "E" => EuclideanSpace ℝ ν |
| local notation "M" => Matrix ν ν ℝ |
| local notation "Y" => EuclideanSpace ℝ ι |
|
|
| lemma Admissible.chart_surgery_impossible {f : E → ι → ℝ} (hf : Admissible f) |
| (hdim : Fintype.card (UpperTriangle ν) = Fintype.card ι) (hn : 1 < Fintype.card ι) |
| {I : Finset ι} (hI : I.Nonempty) [Nonempty ↥(Iᶜ : Finset ι)] |
| (hmax : ∀ u : E, ‖u‖ = 1 → I ⊆ positiveSupport f u → positiveSupport f u = I) |
| {w : E} (hw : ‖w‖ = 1) (hwi : ∀ i ∈ I, 0 < f w i) |
| {r : ι → ℝ} (hr : ∀ i ∈ I, 0 < r i) (hro : ∀ i ∉ I, r i = 0) (hrs : ∑ i, r i = 1) |
| (c : κ → E) (hc : ∀ j, c j ∈ commonPositiveCone f I) (hcn : ∀ j, ‖c j‖ = 1) |
| (hd : Pairwise (fun i j => Disjoint (interior (signChamber (activeVectors f I) (c i))) |
| (interior (signChamber (activeVectors f I) (c j))))) |
| (hcover : orientedLevel f w r ⊆ ⋃ j, interior (signChamber (activeVectors f I) (c j))) |
| {ε : ℝ} (hε : 0 < ε) (g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) |
| (hg : ∀ j, Continuous (g j)) (hgs : ∀ j, ∀ Q ∈ matrixFibre (c j), ∑ i, g j Q i = 1) |
| (hge : ∀ j, ∀ Q ∈ matrixFibre (c j), ‖g j Q - negativeCoordinates f I Q‖ < ε) |
| (z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hz : ∀ i, ε < z i) (hzs : ∑ i, z i = 1) |
| (hgm : ∀ j, ∀ Q ∈ matrixFibre (c j), g j Q ≠ z) : False := by |
| classical |
| let S : κ → Set (traceSphere ν) := fun j => {A | A.val ∈ purityChart f I w (c j)} |
| let U : Set (traceSphere ν) := ⋃ j, S j |
| let C : Set (traceSphere ν) := {A | ∃ p : levelFibre f w r, balancedModel p = A ∧ |
| ∃ t ∈ Icc (0:ℝ) 1, levelHomotopy f I w r c g (t,p) = z} |
| let K : ℝ × traceSphere ν → Y := finiteOpenGlue (fun j => {p | p.2 ∈ S j}) |
| (fun j p => mixedHomotopy f I w (c j) (g j) p.1 p.2.val) |
| let H : traceSphere ν → Y := fun A => WithLp.toLp 2 (signedExtension f A.val) |
| let y : Y := blockVector I (1/2:ℝ) r (1/2:ℝ) z |
| let B : Set Y := {v | ∃ d : ℝ, 0 < d ∧ v = d • y} |
| have hSo : ∀ j, IsOpen (S j) := fun j => |
| hf.purityChart_isOpen I w (c j) Subtype.val continuous_subtype_val (fun A => A.property.1) |
| have hSd : Pairwise (fun i j => Disjoint (S i) (S j)) := by |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun A hA hB => Set.disjoint_left.mp (hd hij) hA.2.2.2.2.2 hB.2.2.2.2.2) |
| have hKd : Pairwise (fun i j => Disjoint {p : ℝ × traceSphere ν | p.2 ∈ S i} |
| {p : ℝ × traceSphere ν | p.2 ∈ S j}) := by |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun p ha hb => Set.disjoint_left.mp (hSd hij) ha hb) |
| have hKe (j : κ) (A : traceSphere ν) (hA : A ∈ S j) (t : ℝ) : |
| K (t,A) = mixedHomotopy f I w (c j) (g j) t A.val := |
| finiteOpenGlue_eq hKd _ hA |
| have hK : ContinuousOn K (Icc (0:ℝ) 1 ×ˢ U) := |
| finiteOpenGlue_continuousOn_prod hSo hSd _ |
| (fun j => hf.mixedHomotopy_continuousOn hI (hc j) (hcn j) (g j) (hg j)) |
| have hC : IsCompact C := hf.level_hits_compact hI c hc hcn hd hcover g hg z |
| have hCU : C ⊆ U := by |
| rintro A ⟨p,rfl,t,ht,he⟩ |
| obtain ⟨j,hj⟩ := mem_iUnion.mp (hcover p.property.1) |
| have hF : fibreHomotopy f I (c j) (g j) t p.val.1 p.val.2 = z := |
| (levelHomotopy_eq hd g hj t).symm.trans he |
| have hunit : IsUnit (rankOne p.val.1 + p.val.2) := by |
| by_contra hnu |
| exact hf.fibreHomotopy_boundary_misses hI (hc j) (hcn j) p.property.1.1 |
| (interior_subset hj) (g j) hε (hge j) z hz ht p.property.2 hnu hF |
| apply mem_iUnion.mpr |
| refine ⟨j,?_⟩ |
| exact balancedFibre_chart p.property.1.1 (hf.orientedLevel_inner_pos hI hw hwi hr p.property.1) |
| (orientedLevel_cone hr p.property.1) hj p.property.2 hunit |
| have hH : Continuous H := (PiLp.continuous_toLp 2 _).comp |
| (hf.signedExtension_continuousOn.comp_continuous continuous_subtype_val (fun A => A.property.1)) |
| have hHz : ∀ A : traceSphere ν, H A ≠ 0 := by |
| intro A h |
| have hzero : signedExtension f A.val = 0 := WithLp.toLp_injective 2 h |
| exact traceSphere_ne_zero A.property ((hf.signedExtension_eq_zero_iff A.property.1).mp hzero) |
| have hK0 : ∀ A ∈ U, K (0,A) = H A := by |
| intro A hA |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hA |
| rw [hKe j A hj 0] |
| exact hf.mixedHomotopy_zero hmax (g j) A.property.1 hj |
| have hKz : ∀ t ∈ Icc (0:ℝ) 1, ∀ A ∈ U, K (t,A) ≠ 0 := by |
| intro t ht A hA |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hA |
| rw [hKe j A hj t] |
| exact hf.mixedHomotopy_nonzero hI (g j) t hj |
| have houtside : ∀ A ∉ U, H A ∉ B := by |
| rintro A hAU ⟨d,hdp,he⟩ |
| obtain ⟨hunit,hline,hAw,hm,hlevel⟩ := hf.original_ray_level hI hmax hw hwi hr hro hrs |
| (fun i => hε.trans (hz i)) hzs A.property hdp he |
| obtain ⟨j,hj⟩ := mem_iUnion.mp (hcover hlevel) |
| exact hAU (mem_iUnion.mpr ⟨j,hunit,hline,hAw,hm,orientedLevel_cone hr hlevel,hj⟩) |
| have hhits : ∀ t ∈ Icc (0:ℝ) 1, ∀ A ∈ U, K (t,A) ∈ B → A ∈ C := by |
| rintro t ht A hAU ⟨d,hdp,he⟩ |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hAU |
| rw [hKe j A hj t] at he |
| obtain ⟨hl,hF,hA⟩ := hf.mixedHomotopy_hit hI hmax (hc j) (hcn j) (g j) (hgs j) |
| hro hrs hzs ht hdp A.property hj he |
| let p : levelFibre f w r := ⟨(lineRepresentative w A.val,negativeFibre A.val),hl,purityChart_fibre hj⟩ |
| refine ⟨p,Subtype.ext hA.symm,t,ht,?_⟩ |
| exact (levelHomotopy_eq hd g hj.2.2.2.2.2 t).trans hF |
| have hend : ∀ A ∈ U, K (1,A) ∉ B := by |
| rintro A hAU ⟨d,hdp,he⟩ |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hAU |
| rw [hKe j A hj 1] at he |
| obtain ⟨-,hF,-⟩ := hf.mixedHomotopy_hit hI hmax (hc j) (hcn j) (g j) (hgs j) |
| hro hrs hzs ⟨zero_le_one,le_rfl⟩ hdp A.property hj he |
| rw [fibreHomotopy_one f I (hcn j) (purityChart_unit hj) (g j) (purityChart_fibre hj)] at hF |
| exact hgm j _ (centeredFibre_mem (hcn j) (purityChart_unit hj) (purityChart_fibre hj)) hF |
| let : CompactSpace (traceSphere ν) := isCompact_iff_compactSpace.mp traceSphere_compact |
| obtain ⟨G,hG,hG0,hGz,hGm⟩ := compact_local_homotopy (isOpen_iUnion hSo) hC hCU H hH hHz |
| K hK hK0 hKz B houtside hhits hend |
| have hy : y ≠ 0 := blockVector_nonzero hI (by norm_num) hr z |
| obtain ⟨A,d,hd,he⟩ := hf.traceHomotopy_ray hdim hn G hG hG0 hGz hy |
| exact hGm A ⟨d,hd,he⟩ |
|
|
| end ChartSurgery |
| end BorsukNine |
|
|
| end OAI |
|
|
