| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.FineCrossMap |
| import OAI.Geometry.Borsuk.Simplicial.BoolCubeConstant |
| import OAI.Geometry.Borsuk.Simplicial.ContigSimplexBoundary |
|
|
| namespace OAI |
|
|
| namespace BorsukNine.SimplicialChains |
| open Finset |
|
|
| noncomputable def approximation (n m r : ℕ) |
| (f : EuclideanSpace ℝ (Fin n) → SignedVertex) : Chain (cross n) →ₗ[F₂] Chain (cross m) := |
| (mapChains (hemisphereComplex (n:ℝ)) (cross m) f).comp (fineCrossMap n r) |
|
|
| def IsFineLabel (n m r : ℕ) (f : EuclideanSpace ℝ (Fin n) → SignedVertex) : Prop := |
| ∀ (s : Finset SignedVertex), s ∈ cross n → |
| ∀ (t : {s : Finset (EuclideanSpace ℝ (Fin n)) // s ∈ hemisphereComplex (n:ℝ)}), |
| t ∈ (fineCrossMap n r (simplex (cross n) s)).support → t.val.image f ∈ cross m |
|
|
| lemma approximation_boundary (n m r : ℕ) (f : EuclideanSpace ℝ (Fin n) → SignedVertex) |
| (hf : IsFineLabel n m r f) : |
| (boundary (cross m)).comp (approximation n m r f) = |
| (approximation n m r f).comp (boundary (cross n)) := by |
| apply Finsupp.lhom_ext |
| intro s a |
| have hh : boundary (cross m) (approximation n m r f (simplex (cross n) s.val)) = |
| approximation n m r f (boundary (cross n) (simplex (cross n) s.val)) := by |
| change boundary (cross m) (mapChains (hemisphereComplex (n:ℝ)) (cross m) f |
| (fineCrossMap n r (simplex (cross n) s.val))) = _ |
| rw [mapChains_boundary_of_support _ (cross m) f _ (hf s.val s.property)] |
| have hb := LinearMap.congr_fun (fineCrossMap_boundary n r) (simplex (cross n) s.val) |
| rw [LinearMap.comp_apply, LinearMap.comp_apply] at hb |
| rw [hb] |
| rfl |
| rw [simplex_of_mem (cross n) s.property] at hh |
| have ha : Finsupp.single s a = a • Finsupp.single s (1:F₂) := by simp |
| simp only [LinearMap.comp_apply, ha, map_smul, hh] |
|
|
| lemma approximation_augmentation (n m r : ℕ) (f : EuclideanSpace ℝ (Fin n) → SignedVertex) |
| (hf : IsFineLabel n m r f) : |
| (augmentation (cross m)).comp (approximation n m r f) = augmentation (cross n) := by |
| apply Finsupp.lhom_ext |
| intro s a |
| have hh : augmentation (cross m) (approximation n m r f (simplex (cross n) s.val)) = |
| augmentation (cross n) (simplex (cross n) s.val) := by |
| change augmentation (cross m) (mapChains (hemisphereComplex (n:ℝ)) (cross m) f |
| (fineCrossMap n r (simplex (cross n) s.val))) = _ |
| rw [mapChains_augmentation_of_support _ (cross m) f _ (hf s.val s.property)] |
| exact LinearMap.congr_fun (fineCrossMap_augmentation n r) _ |
| rw [simplex_of_mem (cross n) s.property] at hh |
| have ha : Finsupp.single s a = a • Finsupp.single s (1:F₂) := by simp |
| simp only [LinearMap.comp_apply, ha, map_smul, hh] |
|
|
| lemma approximation_norm (n m r : ℕ) (hn : 0 < n) |
| (f : EuclideanSpace ℝ (Fin n) → SignedVertex) |
| (hf : ∀ x ∈ annulus n, f (-x) = flip (f x)) : |
| (normMap (cross m) flip).comp (approximation n m r f) = |
| (approximation n m r f).comp (normMap (cross n) flip) := by |
| have hu : (normMap (cross m) flip).comp (mapChains (hemisphereComplex (n:ℝ)) (cross m) f) = |
| (mapChains (hemisphereComplex (n:ℝ)) (cross m) f).comp |
| (normMap (hemisphereComplex (n:ℝ)) (Equiv.neg _)) := by |
| apply mapChains_norm _ (cross m) (Equiv.neg _) flip (hemisphere_neg (n:ℝ)) (cross_flip m) f |
| intro s hs x hx |
| exact hf x (hemisphere_vertex_annulus n hn hs hx) |
| rw [approximation, ← LinearMap.comp_assoc, hu, LinearMap.comp_assoc, |
| fineCrossMap_norm, ← LinearMap.comp_assoc] |
|
|
| lemma approximation_degree (n m r d : ℕ) (f : EuclideanSpace ℝ (Fin n) → SignedVertex) : |
| degreeChains (cross n) d ≤ (degreeChains (cross m) d).comap (approximation n m r f) := by |
| intro c hc |
| exact mapChains_degree_all _ (cross m) f d (fineCrossMap_degree n r d hc) |
|
|
| lemma cross_equivariant_fundamental (n : ℕ) (hn : 1 < n) |
| (T : Chain (cross n) →ₗ[F₂] Chain (cross n)) |
| (hTbd : (boundary (cross n)).comp T = T.comp (boundary (cross n))) |
| (hTN : (normMap (cross n) flip).comp T = T.comp (normMap (cross n) flip)) |
| (hTa : (augmentation (cross n)).comp T = augmentation (cross n)) |
| (hTd : ∀ c ∈ degreeChains (cross n) n, T c ∈ degreeChains (cross n) n) : |
| T (crossFundamental n) = crossFundamental n := by |
| have hc := hTd _ (crossFundamental_degree n (by omega)) |
| have hb : boundary (cross n) (T (crossFundamental n)) = 0 := by |
| rw [← LinearMap.comp_apply, hTbd, LinearMap.comp_apply, |
| crossFundamental_boundary n (by omega), map_zero] |
| have hd : T (crossFundamental n) ⟨crossFacet n (fun _ => false), |
| crossFacet_mem n (by omega) (fun _ => false)⟩ = 1 := by |
| rw [← cross_characteristicDegree_eq_coeff n hn _ hc hb] |
| exact cross_equivariant_degree_one n (by omega) T hTbd hTN hTa |
| rw [cross_top_cycle_eq n hn _ hc hb (fun _ => false), hd, one_smul] |
|
|
| lemma approximation_odd_fundamental (n r : ℕ) (hn : 1 < n) |
| (f : EuclideanSpace ℝ (Fin n) → SignedVertex) (hf : IsFineLabel n n r f) |
| (ho : ∀ x ∈ annulus n, f (-x) = flip (f x)) : |
| approximation n n r f (crossFundamental n) = crossFundamental n := by |
| apply cross_equivariant_fundamental n hn _ (approximation_boundary n n r f hf) |
| (approximation_norm n n r (by omega) f ho) (approximation_augmentation n n r f hf) |
| intro c hc |
| exact approximation_degree n n r n f hc |
|
|
| lemma exists_fine_sphereLabel (n m : ℕ) (hn : 0 < n) [NeZero m] |
| (g : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)) |
| (hg : ContinuousOn g (annulus n)) (hgn : ∀ x ∈ annulus n, ‖g x‖ = 1) : |
| ∃ r : ℕ, IsFineLabel n m r (fun x => sphereLabel (g x)) := by |
| classical |
| have hunif := (annulus_compact n).uniformContinuousOn_of_continuous hg |
| obtain ⟨δ,hδ,hclose⟩ := Metric.uniformContinuousOn_iff.mp hunif ((1:ℝ)/m) |
| (div_pos zero_lt_one (by exact_mod_cast NeZero.pos m)) |
| have hn' : (0:ℝ) < n := by exact_mod_cast hn |
| have hq : ((n:ℝ)-1)/n < 1 := (div_lt_one hn').mpr (by linarith) |
| obtain ⟨r,hr⟩ := exists_pow_lt_of_lt_one (show (0:ℝ) < δ/2 by linarith) hq |
| have hmeshδ : (((n:ℝ)-1)/n)^r * 2 < δ := by linarith |
| refine ⟨r,?_⟩ |
| intro s hs t ht |
| refine ⟨t.property.1.image _, ?_, ?_⟩ |
| · intro v hv |
| obtain ⟨x, _, rfl⟩ := Finset.mem_image.mp hv |
| exact sphereLabel_bound (g x) |
| · intro v hv w hw he |
| obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hv |
| obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hw |
| have hx' := hemisphere_vertex_annulus n hn t.property hx |
| have hy' := hemisphere_vertex_annulus n hn t.property hy |
| apply sphereLabel_compatible (g x) (g y) (hgn x hx') (hgn y hy') _ he |
| exact hclose x hx' y hy' ((fineCrossMap_simplex_mesh n r hs ht x hx y hy).trans_lt hmeshδ) |
|
|
| lemma odd_sphere_simplicial_degree (n : ℕ) (hn : 1 < n) [NeZero n] |
| (g : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n)) |
| (hg : ContinuousOn g (annulus n)) (hgn : ∀ x ∈ annulus n, ‖g x‖ = 1) |
| (hgo : ∀ x ∈ annulus n, g (-x) = -g x) : |
| ∃ r, IsFineLabel n n r (fun x => sphereLabel (g x)) ∧ |
| approximation n n r (fun x => sphereLabel (g x)) (crossFundamental n) = crossFundamental n := by |
| obtain ⟨r,hr⟩ := exists_fine_sphereLabel n n (by omega) g hg hgn |
| refine ⟨r,hr,approximation_odd_fundamental n r hn _ hr ?_⟩ |
| intro x hx |
| rw [hgo x hx, sphereLabel_neg _ (hgn x hx)] |
|
|
| lemma approximation_agree_top (n m r : ℕ) (hn : 0 < n) |
| (f g : EuclideanSpace ℝ (Fin n) → SignedVertex) |
| (hcard : m ≤ n) |
| (hfg : ∀ s ∈ (fineCrossMap n r (crossFundamental n)).support, |
| s.val.image f ∪ s.val.image g ∈ cross m) : |
| approximation n m r f (crossFundamental n) = approximation n m r g (crossFundamental n) := by |
| apply contig_maps_agree_top _ (cross m) f g n _ |
| (fineCrossMap_degree n r n (crossFundamental_degree n hn)) _ hfg |
| (fun s hs => (cross_card m s hs).trans hcard) |
| rw [← LinearMap.comp_apply, fineCrossMap_boundary, LinearMap.comp_apply, |
| crossFundamental_boundary n hn, map_zero] |
|
|
| end BorsukNine.SimplicialChains |
|
|
| end OAI |
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