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0dd3af4 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Configurations.TransitiveInduction
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.PolygonBankEmbedding
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section PolygonAlternatingConfigurations
open Classical CategoryTheory Set
namespace PolygonPlacement
variable {a m n : ℕ}
noncomputable def bankEmbedding (f : Fin n → PolygonPlacement a m)
(hf : Pairwise (fun i j => Apart (f i) (f j))) :
PolygonObject.BankEmbedding (PolygonObject.whole a n) m where
emb := ⟨fun x => bank f hf x.val,fun x y h => Subtype.ext ((bank f hf).injective h)⟩
table := by
choose S hS hcov using fun i => (f i).charts
let c (i : Fin n) (k : Fin m × (CutRing×CutRing) × polygonAlgebra a) :
PolygonObject.Chart (a:=a) n m := ⟨i,k.1,k.2.1,k.2.2⟩
refine ⟨Finset.univ.biUnion (fun i => (S i).image (c i)),?_,?_⟩
· intro k hk
obtain ⟨i,_,hk⟩ := Finset.mem_biUnion.mp hk
obtain ⟨d,hd,rfl⟩ := Finset.mem_image.mp hk
intro z hz
exact ⟨by trivial,hS i d hd z hz⟩
· intro x
obtain ⟨d,hd,hz⟩ := hcov x.val.1 x.val.2
exact ⟨c x.val.1 d,Finset.mem_biUnion.mpr ⟨x.val.1,Finset.mem_univ _,
Finset.mem_image.mpr ⟨d,hd,rfl⟩⟩,rfl,hz⟩
@[simp] theorem bankEmbedding_apply (f : Fin n → PolygonPlacement a m)
(hf : Pairwise (fun i j => Apart (f i) (f j)))
(x : (PolygonObject.whole a n).Point) : bankEmbedding f hf x=f x.val.1 x.val.2 := rfl
theorem alternating_disjoint_configurations (f g : Fin n → PolygonPlacement a m)
(hf : Pairwise (fun i j => Apart (f i) (f j)))
(hg : Pairwise (fun i j => Apart (g i) (g j)))
(hfg : ∀i j,Apart (f i) (g j)) (hm : 3*n+1 < m) :
∃c : polygonAlternatingGroup a m,∀i x,c.val.val (f i x)=g i x := by
let S := Finset.univ.image f ∪ Finset.univ.image g
have hS : S.card≤2*n := by
calc
_≤(Finset.univ.image f).card+(Finset.univ.image g).card := Finset.card_union_le _ _
_≤(Finset.univ : Finset (Fin n)).card+(Finset.univ : Finset (Fin n)).card :=
Nat.add_le_add Finset.card_image_le Finset.card_image_le
_=2*n := by simp; omega
obtain ⟨w,hw,hws⟩ := exists_configuration_avoiding S n (by omega)
have hwf (i j : Fin n) : Apart (w i) (f j) :=
hws i (f j) (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩))
have hwg (i j : Fin n) : Apart (w i) (g j) :=
hws i (g j) (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩))
obtain ⟨c,hc,_⟩ := PolygonObject.BankEmbedding.cycle_three
(bankEmbedding f hf) (bankEmbedding g hg) (bankEmbedding w hw)
(fun p q => (apart_iff _ _).mp (hfg p.val.1 q.val.1) p.val.2 q.val.2)
(fun p q => (apart_iff _ _).mp (hwf q.val.1 p.val.1).symm p.val.2 q.val.2)
(fun p q => (apart_iff _ _).mp (hwg q.val.1 p.val.1).symm p.val.2 q.val.2)
exact ⟨c,fun i x => hc ⟨(i,x),by trivial⟩⟩
theorem alternating_configuration_transitive_sharp (f g : Fin n → PolygonPlacement a m)
(hf : Pairwise (fun i j => Apart (f i) (f j)))
(hg : Pairwise (fun i j => Apart (g i) (g j))) (hm : 3*n+1 < m) :
∃c : polygonAlternatingGroup a m,∀i,c.val • f i=g i := by
let S := Finset.univ.image f ∪ Finset.univ.image g
have hS : S.card≤2*n := by
calc
_≤(Finset.univ.image f).card+(Finset.univ.image g).card := Finset.card_union_le _ _
_≤(Finset.univ : Finset (Fin n)).card+(Finset.univ : Finset (Fin n)).card :=
Nat.add_le_add Finset.card_image_le Finset.card_image_le
_=2*n := by simp; omega
obtain ⟨w,hw,hws⟩ := exists_configuration_avoiding S n (by omega)
have hwf (i j : Fin n) : Apart (w i) (f j) :=
hws i (f j) (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩))
have hwg (i j : Fin n) : Apart (w i) (g j) :=
hws i (g j) (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩))
obtain ⟨u,hu⟩ := alternating_disjoint_configurations f w hf hw (fun i j => (hwf j i).symm) hm
obtain ⟨v,hv⟩ := alternating_disjoint_configurations w g hw hg hwg hm
refine ⟨v*u,fun i => ?_⟩
apply PolygonPlacement.ext
intro x
change v.val.val (u.val.val (f i x))=g i x
rw [hu,hv]
namespace Configuration
theorem alternating_transitive {p : ℕ} (f g : Configuration a m p) (hm : 3*p+1 < m) :
∃c : polygonAlternatingGroup a m,c.val • f=g := by
obtain ⟨c,hc⟩ := alternating_configuration_transitive_sharp f.val g.val f.property g.property hm
exact ⟨c,ext hc⟩
theorem alternating_stabilizer_standard (a p n : ℕ) :
MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)=
(PolygonTracks.bankFixer a p n).comap (polygonAlternatingGroup a (p+n)).subtype := by
ext c
change c.val∈MulAction.stabilizer (polygonFullGroup a (p+n)) (standard a p n) ↔
c.val∈PolygonTracks.bankFixer a p n
rw [stabilizer_standard]
noncomputable def alternatingOrbitHomologyIso (a p n : ℕ) (hm : 3*p+1 < p+n) (q : ℕ) :
groupHomology (Rep.of (TransitiveInduction.permutationRep
(G:=polygonAlternatingGroup a (p+n)) (X:=Configuration a (p+n) p))) q ≅
groupHomology (Rep.trivial ℤ (MulAction.stabilizer (polygonAlternatingGroup a (p+n))
(standard a p n)) ℤ) q :=
TransitiveInduction.homologyIso (standard a p n) (fun f => alternating_transitive _ f hm) q
noncomputable def alternatingRepresentationIso (a m p : ℕ) :
Rep.of (TransitiveInduction.permutationRep
(G:=polygonAlternatingGroup a m) (X:=Configuration a m p)) ≅
Rep.of ((configurationRepresentation a m p).comp (polygonAlternatingGroup a m).subtype) :=
Rep.mkIso
{ toLinearEquiv := chainEquiv a m p
isIntertwining' := fun g => chainEquiv_intertwines g.val }
noncomputable def alternatingColumnHomologyIso (a p n : ℕ) (hm : 3*p+1 < p+n) (q : ℕ) :
groupHomology (Rep.of ((configurationRepresentation a (p+n) p).comp
(polygonAlternatingGroup a (p+n)).subtype)) q ≅
groupHomology (Rep.trivial ℤ (MulAction.stabilizer (polygonAlternatingGroup a (p+n))
(standard a p n)) ℤ) q :=
(groupHomology.functor ℤ _ q).mapIso (alternatingRepresentationIso a (p+n) p).symm ≪≫
alternatingOrbitHomologyIso a p n hm q
end Configuration
end PolygonPlacement
end PolygonAlternatingConfigurations
end SimpleAmenable
end
end
end OAI
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