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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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