| import Mathlib |
| import OAI.Geometry.Borsuk.SphereGeometry.TraceSphere |
| import OAI.Geometry.Borsuk.Simplicial.OddHomotopySurjective |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped MatrixOrder Matrix.Norms.L2Operator |
| open SimplicialChains |
| section TraceHomotopy |
| variable {ν ι : Type*} [Fintype ν] [LinearOrder ν] [Nonempty ν] [Fintype ι] |
| local notation "D" => EuclideanSpace ℝ (UpperTriangle ν) |
| local notation "M" => Matrix ν ν ℝ |
| local notation "Y" => EuclideanSpace ℝ ι |
|
|
| omit [Nonempty ν] in |
| lemma Admissible.traceHomotopy_ray {f : EuclideanSpace ℝ ν → ι → ℝ} (_hf : Admissible f) |
| (hc : Fintype.card (UpperTriangle ν) = Fintype.card ι) (hn : 1 < Fintype.card ι) |
| (G : ℝ × traceSphere ν → Y) |
| (hG : ContinuousOn G (Set.Icc (0:ℝ) 1 ×ˢ Set.univ)) |
| (hG0 : ∀ A : traceSphere ν, G (0,A) = WithLp.toLp 2 (signedExtension f A.val)) |
| (hGz : ∀ t ∈ Set.Icc (0:ℝ) 1, ∀ A : traceSphere ν, G (t,A) ≠ 0) |
| {y : Y} (hy : y ≠ 0) : |
| ∃ A : traceSphere ν, ∃ t : ℝ, 0 < t ∧ G (1,A) = t • y := by |
| classical |
| let n := Fintype.card ι |
| have hnz : n ≠ 0 := by dsimp [n]; omega |
| let : NeZero n := ⟨hnz⟩ |
| let ea : UpperTriangle ν ≃ Fin n := (Fintype.equivFin _).trans (finCongr hc) |
| let eaL := LinearIsometryEquiv.piLpCongrLeft 2 ℝ ℝ ea |
| let ebL := LinearIsometryEquiv.piLpCongrLeft 2 ℝ ℝ (Fintype.equivFin ι) |
| let A : EuclideanSpace ℝ (Fin n) → M := fun x => symmetricCoordinates (eaL.symm x) |
| have hAc : Continuous A := symmetricCoordinates.continuous_of_finiteDimensional.comp eaL.symm.continuous |
| have hAs (x : EuclideanSpace ℝ (Fin n)) : IsSelfAdjoint (A x) := symmetricCoordinates_selfAdjoint _ |
| have hAz (x : EuclideanSpace ℝ (Fin n)) (hx : x ∈ annulus n) : A x ≠ 0 := by |
| intro hz |
| have h0 : eaL.symm x = 0 := symmetricCoordinates_injective (hz.trans (map_zero _).symm) |
| have hx0 : x = 0 := eaL.symm.injective (h0.trans (map_zero _).symm) |
| have hd := hx.1 |
| rw [hx0,norm_zero] at hd |
| norm_num at hd |
| exact hnz hd |
| let N : annulus n → traceSphere ν := fun x => |
| ⟨traceRadial (A x.val),traceRadial_mem (hAs _) (hAz _ x.property)⟩ |
| have hN : Continuous N := by |
| apply Continuous.subtype_mk |
| exact traceRadial_continuousOn.comp_continuous (hAc.comp continuous_subtype_val) |
| (fun x => hAz _ x.property) |
| let H : ℝ × EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n) := fun p => |
| if hx : p.2 ∈ annulus n then ebL (G (p.1,N ⟨p.2,hx⟩)) else 0 |
| have hH : ContinuousOn H (Set.Icc (0:ℝ) 1 ×ˢ annulus n) := by |
| apply continuousOn_iff_continuous_domRestrict.mpr |
| let S := Set.Icc (0:ℝ) 1 ×ˢ annulus n |
| have hcN : Continuous (fun p : S => N ⟨p.val.2,p.property.2⟩) := |
| hN.comp (continuous_subtype_val.snd.subtype_mk _) |
| have hg := hG.comp_continuous (continuous_subtype_val.fst.prodMk hcN) |
| (fun p : S => ⟨p.property.1,Set.mem_univ _⟩) |
| apply (ebL.continuous.comp hg).congr |
| intro p |
| change ebL (G (p.val.1,N ⟨p.val.2,p.property.2⟩)) = H p.val |
| simp only [H,dite_eq_left p.property.2] |
| have hHz : ∀ p ∈ Set.Icc (0:ℝ) 1 ×ˢ annulus n, H p ≠ 0 := by |
| intro p hp hz |
| have he : G (p.1,N ⟨p.2,hp.2⟩) = 0 := ebL.injective (by simpa only [H,dite_eq_left hp.2,map_zero] using hz) |
| exact hGz _ hp.1 _ he |
| have hHo : ∀ x ∈ annulus n, H (0,-x) = -H (0,x) := by |
| intro x hx |
| have hxn : -x ∈ annulus n := by simpa only [annulus,Set.mem_ofPred_eq,norm_neg] using hx |
| simp only [H,dite_eq_left hx,dite_eq_left hxn,hG0,N] |
| have hAn : A (-x) = -A x := by simp only [A,map_neg] |
| rw [hAn,traceRadial_neg,signedExtension_odd,WithLp.toLp_neg,map_neg] |
| have hY : ‖ebL (unitRadial y)‖ = 1 := by rw [ebL.norm_map,unitRadial_norm hy] |
| obtain ⟨x,hx,he⟩ := odd_homotopy_surjective n hn H hH hHz hHo hY |
| let z : Y := G (1,N ⟨x,hx⟩) |
| have hz : z ≠ 0 := hGz _ ⟨zero_le_one,le_rfl⟩ _ |
| have he' : unitRadial z = unitRadial y := by |
| apply ebL.injective |
| have heL : ebL (unitRadial z) = unitRadial (ebL z) := by simp only [unitRadial,map_smul,ebL.norm_map] |
| rw [heL] |
| simpa only [H,dite_eq_left hx] using he |
| refine ⟨N ⟨x,hx⟩,‖z‖ / ‖y‖,div_pos (norm_pos_iff.mpr hz) (norm_pos_iff.mpr hy),?_⟩ |
| have hh := congrArg (fun v : Y => ‖z‖ • v) he' |
| simpa only [unitRadial,smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hz),one_smul, |
| ← div_eq_mul_inv] using hh |
|
|
| end TraceHomotopy |
| end BorsukNine |
|
|
| end OAI |
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