| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.LinearMapSupport |
| import OAI.Geometry.Borsuk.Simplicial.DoubleEraseSumFun |
|
|
| namespace OAI |
|
|
| namespace BorsukNine.SimplicialChains |
| open Finset |
| variable {V W : Type*} [DecidableEq V] [DecidableEq W] |
|
|
| lemma contigSimplex_boundary (L : PreAbstractSimplicialComplex W) (f g : V → W) |
| {s : Finset V} (hs : s.image f ∪ s.image g ∈ L) : |
| boundary L (contigSimplex L f g s) = |
| mapSimplex L f s + mapSimplex L g s + ∑ a ∈ s, contigSimplex L f g (s.erase a) := by |
| induction s using Finset.strongInductionOn with |
| | _ s ih => |
| have hn : s.Nonempty := by |
| by_contra h |
| have he := Finset.not_nonempty_iff_eq_empty.mp h |
| have hs' : (∅ : Finset W) ∈ L := by simpa [he] using hs |
| simpa using (L.isRelLowerSet_faces hs').1 |
| have hf : s.image f ∈ L := (L.isRelLowerSet_faces hs).2 Finset.subset_union_left (hn.image f) |
| have hg : s.image g ∈ L := (L.isRelLowerSet_faces hs).2 Finset.subset_union_right (hn.image g) |
| have hb : f hn.choose ∈ s.image f ∪ s.image g := |
| Finset.mem_union_left _ (Finset.mem_image_of_mem f hn.choose_spec) |
| let z := mapSimplex L f s + mapSimplex L g s + ∑ a ∈ s, contigSimplex L f g (s.erase a) |
| have hi (a : V) (ha : a ∈ s) : boundary L (contigSimplex L f g (s.erase a)) = |
| mapSimplex L f (s.erase a) + mapSimplex L g (s.erase a) + |
| ∑ b ∈ s.erase a, contigSimplex L f g ((s.erase a).erase b) := by |
| rcases (s.erase a).eq_empty_or_nonempty with he | hne |
| · simp [he, mapSimplex] |
| · apply ih (s.erase a) (Finset.erase_ssubset ha) |
| apply (L.isRelLowerSet_faces hs).2 |
| · exact Finset.union_subset_union (Finset.image_subset_image (Finset.erase_subset a s)) |
| (Finset.image_subset_image (Finset.erase_subset a s)) |
| · exact (hne.image f).mono Finset.subset_union_left |
| have hzbd : boundary L z = 0 := by |
| simp only [z, map_add, map_sum, mapSimplex_boundary L f hf, mapSimplex_boundary L g hg] |
| rw [Finset.sum_congr rfl hi] |
| simp only [Finset.sum_add_distrib] |
| rw [double_erase_sum_fun, add_zero] |
| exact chain_add_self L _ |
| have hzaug : augmentation L z = 0 := by |
| simp only [z, map_add, map_sum, mapSimplex_augmentation L f hf, |
| mapSimplex_augmentation L g hg, contigSimplex_augmentation, Finset.sum_const_zero, add_zero] |
| exact CharTwo.add_self_eq_zero _ |
| have hzcone : ∀ t ∈ z.support, insert (f hn.choose) t.val ∈ L := by |
| intro t ht |
| apply (L.isRelLowerSet_faces hs).2 _ (Finset.insert_nonempty _ _) |
| apply Finset.insert_subset hb |
| rcases Finset.mem_union.mp (Finsupp.support_add ht) with ht | ht |
| · rcases Finset.mem_union.mp (Finsupp.support_add ht) with htf | htg |
| · rw [support_mapSimplex L f s htf]; exact Finset.subset_union_left |
| · rw [support_mapSimplex L g s htg]; exact Finset.subset_union_right |
| · obtain ⟨a,ha,hta⟩ := support_sum_chains L s _ ht |
| exact (contigSimplex_carrier L f g (s.erase a) hta).trans |
| (Finset.union_subset_union (Finset.image_subset_image (Finset.erase_subset a s)) |
| (Finset.image_subset_image (Finset.erase_subset a s))) |
| rw [contigSimplex_nonempty L f g hn] |
| change boundary L (cone L (f hn.choose) z) = z |
| rw [cone_boundary L _ z hzcone, hzbd, hzaug, map_zero, zero_smul, add_zero, add_zero] |
|
|
| noncomputable def contigMap (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) : Chain K →ₗ[F₂] Chain L := |
| Finsupp.linearCombination F₂ (fun s => contigSimplex L f g s.val) |
|
|
| lemma contigMap_simplex (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) {s : Finset V} (hs : s ∈ K) : |
| contigMap K L f g (simplex K s) = contigSimplex L f g s := by |
| simp [simplex_of_mem K hs, contigMap, Finsupp.linearCombination_single] |
|
|
| lemma contigMap_simplex_subset (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) {s t : Finset V} |
| (ht : t ∈ K) (hs : s ⊆ t) : |
| contigMap K L f g (simplex K s) = contigSimplex L f g s := by |
| rcases s.eq_empty_or_nonempty with rfl | hn |
| · simp |
| · exact contigMap_simplex K L f g ((K.isRelLowerSet_faces ht).2 hs hn) |
|
|
| lemma contigMap_degree (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) (d : ℕ) : |
| degreeChains K d ≤ (degreeChains L (d+1)).comap (contigMap K L f g) := by |
| apply grade_map K L _ d (d+1) |
| intro s hs hd |
| rw [contigMap_simplex K L f g hs, ← hd] |
| exact contigSimplex_degree L f g s |
|
|
| lemma contigMap_boundary_of_support (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) (c : Chain K) |
| (hc : ∀ s ∈ c.support, s.val.image f ∪ s.val.image g ∈ L) : |
| boundary L (contigMap K L f g c) = mapChains K L f c + mapChains K L g c + |
| contigMap K L f g (boundary K c) := by |
| let A : Chain K →ₗ[F₂] Chain L := |
| mapChains K L f + mapChains K L g + (contigMap K L f g).comp (boundary K) |
| change ((boundary L).comp (contigMap K L f g)) c = A c |
| apply linearMap_eq_of_support |
| intro s hs |
| simp only [LinearMap.comp_apply, A, LinearMap.add_apply] |
| rw [contigMap_simplex K L f g s.property, contigSimplex_boundary L f g (hc s hs), |
| mapChains_simplex K L f s.property, mapChains_simplex K L g s.property, |
| boundary_simplex K s.property, map_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro a _ |
| exact (contigMap_simplex_subset K L f g s.property (Finset.erase_subset a s.val)).symm |
|
|
| lemma contig_maps_agree_top (K : PreAbstractSimplicialComplex V) |
| (L : PreAbstractSimplicialComplex W) (f g : V → W) (d : ℕ) (c : Chain K) |
| (hc : c ∈ degreeChains K d) (hbd : boundary K c = 0) |
| (hfg : ∀ s ∈ c.support, s.val.image f ∪ s.val.image g ∈ L) |
| (hL : ∀ s ∈ L, s.card ≤ d) : mapChains K L f c = mapChains K L g c := by |
| have hh : contigMap K L f g c = 0 := by |
| have ht := contigMap_degree K L f g d hc |
| rw [degreeChains_eq_bot_of_card_bound L (d+1) (fun s hs => by have := hL s hs; omega)] at ht |
| exact ht |
| have he := contigMap_boundary_of_support K L f g c hfg |
| rw [hh, map_zero, hbd, map_zero, add_zero] at he |
| have h := eq_neg_of_add_eq_zero_left he.symm |
| have hn : -(mapChains K L g c) = mapChains K L g c := by |
| ext s |
| exact CharTwo.neg_eq _ |
| exact h.trans hn |
|
|
| end BorsukNine.SimplicialChains |
|
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| end OAI |
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