| import Mathlib | |
| import OAI.Geometry.Borsuk.Simplicial.MapChainsEquivSimplexAll | |
| namespace OAI | |
| namespace BorsukNine.SimplicialChains | |
| open Finset | |
| variable {V W : Type*} [DecidableEq V] [DecidableEq W] | |
| omit [DecidableEq V] in | |
| lemma linearMap_eq_of_support {K : PreAbstractSimplicialComplex V} | |
| {M : Type*} [AddCommMonoid M] [Module F₂ M] | |
| (T U : Chain K →ₗ[F₂] M) (c : Chain K) | |
| (hc : ∀ s ∈ c.support, T (simplex K s.val) = U (simplex K s.val)) : T c = U c := by | |
| rw [chain_eq_sum K c, map_sum, map_sum] | |
| apply Finset.sum_congr rfl | |
| intro s hs | |
| rw [map_smul, map_smul, hc s hs] | |
| lemma mapChains_boundary_simplex_of_image (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (f : V → W) | |
| {s : Finset V} (hs : s ∈ K) (ht : s.image f ∈ L) : | |
| boundary L (mapChains K L f (simplex K s)) = | |
| mapChains K L f (boundary K (simplex K s)) := by | |
| rw [mapChains_simplex K L f hs, boundary_simplex K hs, map_sum] | |
| have h (a : V) : mapChains K L f (simplex K (s.erase a)) = mapSimplex L f (s.erase a) := | |
| mapChains_simplex_subset K L f hs (Finset.erase_subset a s) | |
| simp_rw [h] | |
| by_cases hinj : Set.InjOn f s | |
| · rw [mapSimplex, ite_eq_left (Finset.card_image_iff.mpr hinj), boundary_simplex L ht] | |
| rw [Finset.sum_image] | |
| · apply Finset.sum_congr rfl | |
| intro a ha | |
| have hj : Set.InjOn f (s.erase a) := hinj.mono (Finset.erase_subset a s) | |
| rw [mapSimplex, ite_eq_left (Finset.card_image_iff.mpr hj), image_erase_of_injOn f s hinj ha] | |
| · exact hinj | |
| · rw [mapSimplex, ite_eq_right (mt Finset.card_image_iff.mp hinj), map_zero, | |
| mapSimplex_boundary_degenerate L f s hinj] | |
| lemma mapChains_boundary_of_support (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (f : V → W) (c : Chain K) | |
| (hc : ∀ s ∈ c.support, s.val.image f ∈ L) : | |
| boundary L (mapChains K L f c) = mapChains K L f (boundary K c) := by | |
| apply linearMap_eq_of_support ((boundary L).comp (mapChains K L f)) | |
| ((mapChains K L f).comp (boundary K)) c | |
| intro s hs | |
| exact mapChains_boundary_simplex_of_image K L f s.property (hc s hs) | |
| omit [DecidableEq V] in | |
| lemma mapChains_augmentation_simplex_of_image (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (f : V → W) | |
| {s : Finset V} (hs : s ∈ K) (ht : s.image f ∈ L) : | |
| augmentation L (mapChains K L f (simplex K s)) = augmentation K (simplex K s) := by | |
| rw [mapChains_simplex K L f hs, augmentation_simplex K hs] | |
| by_cases hcard : (s.image f).card = s.card | |
| · rw [mapSimplex, ite_eq_left hcard, augmentation_simplex L ht, hcard] | |
| · have hne : s.card ≠ 1 := by | |
| intro hc | |
| obtain ⟨v,rfl⟩ := Finset.card_eq_one.mp hc | |
| simp at hcard | |
| simp [mapSimplex, hcard, hne] | |
| omit [DecidableEq V] in | |
| lemma mapChains_augmentation_of_support (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (f : V → W) (c : Chain K) | |
| (hc : ∀ s ∈ c.support, s.val.image f ∈ L) : | |
| augmentation L (mapChains K L f c) = augmentation K c := by | |
| apply linearMap_eq_of_support ((augmentation L).comp (mapChains K L f)) (augmentation K) c | |
| intro s hs | |
| exact mapChains_augmentation_simplex_of_image K L f s.property (hc s hs) | |
| omit [DecidableEq V] in | |
| lemma mapChains_degree_all (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (f : V → W) (d : ℕ) : | |
| degreeChains K d ≤ (degreeChains L d).comap (mapChains K L f) := by | |
| apply grade_map K L _ d d | |
| intro s hs hd | |
| rw [mapChains_simplex K L f hs, mapSimplex] | |
| split_ifs with hh | |
| · rw [← hd, ← hh] | |
| exact simplex_mem_degree_all L _ | |
| · exact Submodule.zero_mem _ | |
| lemma mapChains_equivariant (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (e : V ≃ V) (d : W ≃ W) | |
| (he : ∀ s, s ∈ K ↔ s.image e ∈ K) (hd : ∀ s, s ∈ L ↔ s.image d ∈ L) | |
| (f : V → W) (hf : ∀ s ∈ K, ∀ x ∈ s, f (e x) = d (f x)) : | |
| (mapChains L L d).comp (mapChains K L f) = | |
| (mapChains K L f).comp (mapChains K K e) := by | |
| apply Finsupp.lhom_ext | |
| intro s r | |
| have him : (s.val.image e).image f = (s.val.image f).image d := by | |
| simp only [Finset.image_image] | |
| exact Finset.image_congr (fun x hx => hf s.val s.property x hx) | |
| have hcard : (s.val.image e).card = s.val.card := Finset.card_image_iff.mpr e.injective.injOn | |
| have hcard' : ((s.val.image f).image d).card = (s.val.image f).card := | |
| Finset.card_image_iff.mpr d.injective.injOn | |
| have hh : mapChains L L d (mapChains K L f (simplex K s.val)) = | |
| mapChains K L f (mapChains K K e (simplex K s.val)) := by | |
| rw [mapChains_simplex K L f s.property, mapChains_equiv_simplex_all K e he, | |
| mapChains_simplex K L f ((he s.val).mp s.property)] | |
| simp only [mapSimplex, him, hcard, hcard'] | |
| split_ifs | |
| · exact mapChains_equiv_simplex_all L d hd _ | |
| · exact map_zero _ | |
| rw [simplex_of_mem K s.property] at hh | |
| have hr : Finsupp.single s r = r • Finsupp.single s (1 : F₂) := by simp | |
| simp only [LinearMap.comp_apply, hr, map_smul, hh] | |
| lemma mapChains_norm (K : PreAbstractSimplicialComplex V) | |
| (L : PreAbstractSimplicialComplex W) (e : V ≃ V) (d : W ≃ W) | |
| (he : ∀ s, s ∈ K ↔ s.image e ∈ K) (hd : ∀ s, s ∈ L ↔ s.image d ∈ L) | |
| (f : V → W) (hf : ∀ s ∈ K, ∀ x ∈ s, f (e x) = d (f x)) : | |
| (normMap L d).comp (mapChains K L f) = | |
| (mapChains K L f).comp (normMap K e) := by | |
| simp only [normMap, LinearMap.add_comp, LinearMap.comp_add, LinearMap.id_comp, | |
| LinearMap.comp_id, mapChains_equivariant K L e d he hd f hf] | |
| end BorsukNine.SimplicialChains | |
| end OAI | |