| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.Supports |
| import OAI.Geometry.Borsuk.Simplicial.NullHomotopyLocalModels |
|
|
| namespace OAI |
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|
| open Set MeasureTheory |
| namespace BorsukNine |
| open Set SimplicialChains |
| noncomputable def normSphere (n : ℕ) : Set (EuclideanSpace ℝ (Fin n)) := {u | ‖u‖ = 1} |
| section SphereNullModels |
| variable {n : ℕ} {κ : Type*} [Fintype κ] [DecidableEq κ] |
| local notation "E" => EuclideanSpace ℝ (Fin n) |
| local notation "S" => normSphere n |
| lemma sphere_null_local_models (hn : 1 < n) |
| (H C : ℝ × S → E) |
| (hH : ContinuousOn H (Icc (0:ℝ) 1 ×ˢ univ)) |
| (hHz : ∀ t ∈ Icc (0:ℝ) 1, ∀ u : S, H (t,u) ≠ 0) |
| (hC : ContinuousOn C (Icc (0:ℝ) 1 ×ˢ univ)) |
| (hCz : ∀ t ∈ Icc (0:ℝ) 1, ∀ u : S, C (t,u) ≠ 0) |
| (hC1 : ∀ u, C (1,u) = H (0,u)) |
| (G : κ → S → E) (hG : ∀ a, Continuous (G a)) (hGz : ∀ a u, G a u ≠ 0) |
| (hGo : ∀ a (u v : S), v.val = -u.val → G a v = -G a u) |
| (U : κ → Set S) (hU : ∀ a, IsOpen (U a)) |
| (heq : ∀ a u, u ∈ U a → H (1,u) = G a u) |
| {y : E} (hy : y ≠ 0) |
| (hmiss : ∀ u : S, ¬ ∃ d : ℝ, 0 < d ∧ C (0,u) = d • y) |
| (hcover : ∀ u : S, (∃ d : ℝ, 0 < d ∧ H (1,u) = d • y) → ∃ a, u ∈ U a) |
| (hroot : ∀ a u, (∃ d : ℝ, 0 < d ∧ G a u = d • y) → u ∈ U a) |
| (hdis : Pairwise (fun a b => Disjoint (U a) (U b))) : (Fintype.card κ : F₂) = 0 := by |
| classical |
| let : NeZero n := ⟨by omega⟩ |
| let b : Fin n → Bool := fun _ => false |
| let L := Submodule.reflection (ℝ ∙ (unitRadial y - facetCenter n b))ᗮ |
| have hRy : L (unitRadial y) = facetCenter n b := |
| Submodule.reflection_sub ((unitRadial_norm hy).trans (facetCenter_norm n b).symm) |
| have hray (v : E) (hv : v ≠ 0) (he : unitRadial (L v) = facetCenter n b) : |
| ∃ d : ℝ, 0 < d ∧ v = d • y := by |
| have hu : unitRadial v = unitRadial y := by |
| apply L.injective |
| rw [hRy] |
| simpa only [unitRadial,map_smul,L.norm_map] using he |
| refine ⟨‖v‖ / ‖y‖,div_pos (norm_pos_iff.mpr hv) (norm_pos_iff.mpr hy),?_⟩ |
| have hh := congrArg (fun z : E => ‖v‖ • z) hu |
| simpa only [unitRadial,smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hv),one_smul,div_eq_mul_inv] using hh |
| let N : annulus n → S := fun x => ⟨unitRadial x.val,unitRadial_norm (annulus_ne_zero n (by omega) x.property)⟩ |
| have hN : Continuous N := by |
| apply Continuous.subtype_mk |
| exact (unitRadial_continuousOn (fun _ hx => annulus_ne_zero n (by omega) hx)).domRestrict |
| have hNneg (x) (hx : x ∈ annulus n) (hxn : -x ∈ annulus n) : (N ⟨-x,hxn⟩).val = -(N ⟨x,hx⟩).val := unitRadial_neg x |
| let extH (P : ℝ × S → E) : ℝ × E → E := fun p => if hx : p.2 ∈ annulus n then L (P (p.1,N ⟨p.2,hx⟩)) else 0 |
| have hext (P : ℝ × S → E) (hP : ContinuousOn P (Icc (0:ℝ) 1 ×ˢ univ)) : |
| ContinuousOn (extH P) (Icc (0:ℝ) 1 ×ˢ annulus n) := by |
| apply continuousOn_iff_continuous_domRestrict.mpr |
| let Q := Icc (0:ℝ) 1 ×ˢ annulus n |
| have hcN : Continuous (fun p : Q => N ⟨p.val.2,p.property.2⟩) := hN.comp (continuous_subtype_val.snd.subtype_mk _) |
| have hg := hP.comp_continuous (continuous_subtype_val.fst.prodMk hcN) (fun p : Q => ⟨p.property.1,mem_univ _⟩) |
| apply (L.continuous.comp hg).congr |
| intro p |
| simp only [Set.domRestrict_apply,Function.comp_apply,extH,dite_eq_left p.property.2] |
| have hextz (P : ℝ × S → E) (hPz : ∀ t ∈ Icc (0:ℝ) 1, ∀ u, P (t,u) ≠ 0) : |
| ∀ p ∈ Icc (0:ℝ) 1 ×ˢ annulus n, extH P p ≠ 0 := by |
| intro p hp hz |
| exact hPz _ hp.1 _ (L.injective (by simpa only [extH,dite_eq_left hp.2,map_zero] using hz)) |
| let G' : κ → E → E := fun a x => if hx : x ∈ annulus n then L (G a (N ⟨x,hx⟩)) else 0 |
| have hG' (a) : ContinuousOn (G' a) (annulus n) := by |
| apply continuousOn_iff_continuous_domRestrict.mpr |
| apply (L.continuous.comp ((hG a).comp hN)).congr |
| intro x |
| simp only [Set.domRestrict_apply,Function.comp_apply,G',dite_eq_left x.property] |
| have hGz' (a x) (hx : x ∈ annulus n) : G' a x ≠ 0 := by |
| intro hz |
| exact hGz a _ (L.injective (by simpa only [G',dite_eq_left hx,map_zero] using hz)) |
| have hGo' (a x) (hx : x ∈ annulus n) : G' a (-x) = -G' a x := by |
| have hxn : -x ∈ annulus n := by simpa only [annulus,Set.mem_ofPred_eq,norm_neg] using hx |
| simp only [G',dite_eq_left hx,dite_eq_left hxn,hGo a _ _ (hNneg x hx hxn),map_neg] |
| refine null_homotopy_local_models n hn b (extH H) (hext H hH) (hextz H hHz) (extH C) (hext C hC) (hextz C hCz) |
| (fun x => by simp only [extH,hC1]) ?_ G' hG' hGz' hGo' |
| (fun a => N ⁻¹' U a) (fun a => (hU a).preimage hN) ?_ ?_ ?_ ?_ |
| · intro x hx he |
| apply hmiss (N ⟨x,hx⟩) |
| apply hray _ (hCz _ ⟨le_rfl,zero_le_one⟩ _) |
| simpa only [extH,dite_eq_left hx] using he |
| · intro a x hx |
| simp only [extH,G',dite_eq_left x.property,heq a _ hx] |
| · intro x hx |
| apply hcover (N x) |
| apply hray _ (hHz _ ⟨zero_le_one,le_rfl⟩ _) |
| simpa only [extH,dite_eq_left x.property] using hx |
| · intro a x hx |
| apply hroot a (N x) |
| apply hray _ (hGz a _) |
| simpa only [G',dite_eq_left x.property] using hx |
| · intro a a' haa |
| exact (hdis haa).preimage N |
| end SphereNullModels |
| end BorsukNine |
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| end OAI |
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