| import Mathlib | |
| import OAI.Geometry.Borsuk.Simplicial.FineCrossMap | |
| namespace OAI | |
| namespace BorsukNine.SimplicialChains | |
| open Finset | |
| section GeneralTarget | |
| variable {W : Type*} [DecidableEq W] | |
| noncomputable def fineMap (n r : ℕ) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) : Chain (cross n) →ₗ[F₂] Chain L := | |
| (mapChains (hemisphereComplex (n:ℝ)) L f).comp (fineCrossMap n r) | |
| def FineTo (n r : ℕ) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) : Prop := | |
| ∀ s ∈ cross n, ∀ t ∈ (fineCrossMap n r (simplex (cross n) s)).support, | |
| t.val.image f ∈ L | |
| lemma fineMap_boundary (n r : ℕ) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) (hf : FineTo n r L f) : | |
| (boundary L).comp (fineMap n r L f) = (fineMap n r L f).comp (boundary (cross n)) := by | |
| apply Finsupp.lhom_ext | |
| intro s a | |
| have hh : boundary L (fineMap n r L f (simplex (cross n) s.val)) = | |
| fineMap n r L f (boundary (cross n) (simplex (cross n) s.val)) := by | |
| change boundary L (mapChains (hemisphereComplex (n:ℝ)) L f | |
| (fineCrossMap n r (simplex (cross n) s.val))) = _ | |
| rw [mapChains_boundary_of_support _ L 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 fineMap_degree (n r d : ℕ) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) : | |
| degreeChains (cross n) d ≤ (degreeChains L d).comap (fineMap n r L f) := by | |
| intro c hc | |
| exact mapChains_degree_all _ L f d (fineCrossMap_degree n r d hc) | |
| lemma mapChains_id (L : PreAbstractSimplicialComplex W) : | |
| mapChains L L (Equiv.refl W) = LinearMap.id := by | |
| apply Finsupp.lhom_ext | |
| intro s a | |
| have hh : mapChains L L (Equiv.refl W) (simplex L s.val) = simplex L s.val := by | |
| simpa only [Equiv.coe_refl, Finset.image_id] using | |
| mapChains_equiv_simplex_all L (Equiv.refl W) (by simp) s.val | |
| rw [simplex_of_mem L s.property] at hh | |
| have ha : Finsupp.single s a = a • Finsupp.single s (1:F₂) := by simp | |
| simp only [ha, map_smul, hh, LinearMap.id_apply] | |
| lemma normMap_id (L : PreAbstractSimplicialComplex W) : normMap L (Equiv.refl W) = 0 := by | |
| rw [normMap, mapChains_id] | |
| ext c s | |
| exact CharTwo.add_self_eq_zero _ | |
| lemma fineMap_even_norm (n r : ℕ) (hn : 0 < n) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) (hf : ∀ x ∈ annulus n, f (-x) = f x) : | |
| (fineMap n r L f).comp (normMap (cross n) flip) = 0 := by | |
| have hu : (normMap L (Equiv.refl W)).comp (mapChains (hemisphereComplex (n:ℝ)) L f) = | |
| (mapChains (hemisphereComplex (n:ℝ)) L f).comp | |
| (normMap (hemisphereComplex (n:ℝ)) (Equiv.neg _)) := by | |
| apply mapChains_norm _ L (Equiv.neg _) (Equiv.refl W) (hemisphere_neg (n:ℝ)) | |
| (by simp) f | |
| intro s hs x hx | |
| exact hf x (hemisphere_vertex_annulus n hn hs hx) | |
| rw [normMap_id, LinearMap.zero_comp] at hu | |
| rw [fineMap, LinearMap.comp_assoc, ← fineCrossMap_norm, ← LinearMap.comp_assoc, | |
| ← hu, LinearMap.zero_comp] | |
| noncomputable def projectiveCycle (n r : ℕ) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) : Chain L := | |
| fineMap n r L f (crossTower n (n-1)) | |
| lemma projectiveCycle_degree (n r : ℕ) (hn : 0 < n) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) : projectiveCycle n r L f ∈ degreeChains L n := by | |
| have hc := crossTower_degree n (n-1) | |
| rw [Nat.sub_add_cancel hn] at hc | |
| exact fineMap_degree n r n L f hc | |
| lemma projectiveCycle_boundary (n r : ℕ) (hn : 1 < n) (L : PreAbstractSimplicialComplex W) | |
| (f : EuclideanSpace ℝ (Fin n) → W) (hf : FineTo n r L f) | |
| (he : ∀ x ∈ annulus n, f (-x) = f x) : boundary L (projectiveCycle n r L f) = 0 := by | |
| unfold projectiveCycle | |
| rw [← LinearMap.comp_apply, fineMap_boundary n r L f hf, LinearMap.comp_apply] | |
| have heq : n-1 = (n-2)+1 := by omega | |
| rw [heq, crossTower_boundary n (n-2) (by omega), ← LinearMap.comp_apply, | |
| fineMap_even_norm n r (by omega) L f he, LinearMap.zero_apply] | |
| end GeneralTarget | |
| end BorsukNine.SimplicialChains | |
| end OAI | |