File size: 6,940 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonAlternatingStabilization
import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonAlternatingConfigurations
import OAI.GroupTheory.SimpleAmenable.Configurations.StabilizationComposition
import OAI.GroupTheory.SimpleAmenable.Homology.PermutationAugmentation
namespace OAI
section
section
open Classical CategoryTheory CategoryTheory.Limits Representation Rep Finsupp
namespace SimpleAmenable
attribute [local instance 1200] Rep.hV2
namespace PolygonPlacement.Configuration
variable {a m p n : ℕ}
lemma stabilizeAlternating_smul_standard (g : polygonAlternatingGroup a n) :
PolygonTracks.stabilizeAlternating a p n g • standard a p n=standard a p n := by
apply ext
intro i
apply PolygonPlacement.ext
intro x
exact PolygonTracks.stabilize_left g.val (i,x)
noncomputable def alternatingRestrict (a p n : ℕ) (hn : 32 ≤ n)
(g : MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)) :
polygonAlternatingGroup a n := by
have hg : g.val.val∈PolygonTracks.bankFixer a p n := by
have hh := g.property
exact (show g.val ∈ (PolygonTracks.bankFixer a p n).comap
(polygonAlternatingGroup a (p+n)).subtype from
(alternating_stabilizer_standard a p n) ▸ hh)
refine ⟨PolygonTracks.restrict ⟨g.val.val,hg⟩,?_⟩
rw [← PolygonTracks.stabilize_alternating_comap a p n hn]
change PolygonTracks.stabilize a p n (PolygonTracks.restrict _)∈polygonAlternatingGroup a (p+n)
rw [PolygonTracks.stabilize_restrict]
exact g.val.property
noncomputable def alternatingStabilizerEquiv (a p n : ℕ) (hn : 32 ≤ n) :
polygonAlternatingGroup a n ≃*
MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n) where
toFun g := ⟨PolygonTracks.stabilizeAlternating a p n g,stabilizeAlternating_smul_standard g⟩
invFun := alternatingRestrict a p n hn
left_inv g := by
apply Subtype.ext
change PolygonTracks.restrict ⟨PolygonTracks.stabilize a p n g.val,_⟩=g.val
exact PolygonTracks.restrict_stabilize g.val
right_inv g := by
apply Subtype.ext
apply Subtype.ext
change PolygonTracks.stabilize a p n (PolygonTracks.restrict _) = g.val.val
exact PolygonTracks.stabilize_restrict _
map_mul' g h := Subtype.ext ((PolygonTracks.stabilizeAlternating a p n).map_mul g h)
lemma alternatingStabilizerEquiv_subtype (a p n : ℕ) (hn : 32 ≤ n) :
(MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)).subtype.comp
(alternatingStabilizerEquiv a p n hn).toMonoidHom=PolygonTracks.stabilizeAlternating a p n := rfl
lemma alternatingStabilizer_perfect (a p n : ℕ) (hn : 32 ≤ n) :
Group.IsPerfect (MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)) := by
have := polygonAlternatingGroup_perfect a n (by omega)
exact Group.IsPerfect.ofSurjective (f:=(alternatingStabilizerEquiv a p n hn).toMonoidHom)
(alternatingStabilizerEquiv a p n hn).surjective
noncomputable abbrev altColumn (a m p : ℕ) :=
Rep.res (polygonAlternatingGroup a m).subtype (column a m p)
noncomputable abbrev altFaceHom (a m p : ℕ) (i : Fin (p+1)) :
altColumn a m (p+1) ⟶ altColumn a m p :=
(Rep.resFunctor (polygonAlternatingGroup a m).subtype).map (faceHom a m p i)
noncomputable abbrev altBoundary (a m p : ℕ) : altColumn a m (p+1) ⟶ altColumn a m p :=
(Rep.resFunctor (polygonAlternatingGroup a m).subtype).map (boundary a m p)
lemma altBoundary_square (a m p : ℕ) : altBoundary a m (p+1) ≫ altBoundary a m p=0 := by
rw [← Functor.map_comp, boundary_square, Functor.map_zero]
lemma altBoundary_exact (a m p : ℕ) (hm : 33 ≤ m) (hp : p ≤ 2) :
(ShortComplex.mk (altBoundary a m (p+1)) (altBoundary a m p) (altBoundary_square a m p)).Exact := by
exact (Rep.res_map_exact (polygonAlternatingGroup a m).subtype
(ShortComplex.mk (boundary a m (p+1)) (boundary a m p) (boundary_square a m p))).mpr
(boundary_exact a m p hm hp)
lemma altBoundary_zero_epi (a m : ℕ) (hm : 0 < m) : Epi (altBoundary a m 0) := by
rw [Rep.epi_iff_surjective]
exact (Rep.epi_iff_surjective (boundary a m 0)).mp (boundary_zero_epi a m hm)
lemma altFace_H0_eq (a m p : ℕ) (hm : 3*p+1 < m) (i j : Fin (p+1)) :
(groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p i)=
(groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p j) := by
let f := standardLE a m p (by omega : p ≤ m)
exact TransitiveInduction.map_equivariant_H0_eq (G:=polygonAlternatingGroup a m)
(face i) (face j) (fun g x => face_smul i g.val x) (fun g x => face_smul j g.val x)
f (fun g => alternating_transitive f g hm)
lemma altFace_H0_isIso (a m p : ℕ) (hm : 3*(p+1)+1 < m) (i : Fin (p+1)) :
IsIso ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p i)) := by
let f := standardLE a m (p+1) (by omega : p+1 ≤ m)
exact TransitiveInduction.isIso_map_equivariant_H0 (G:=polygonAlternatingGroup a m)
(face i) (fun g x => face_smul i g.val x) f
(fun g => alternating_transitive f g hm) (fun g => alternating_transitive _ g (by omega))
lemma altBoundary_two_H0_isIso (a m : ℕ) (hm : 10 < m) :
IsIso ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altBoundary a m 2)) := by
have he : altBoundary a m 2=altFaceHom a m 2 0-altFaceHom a m 2 1+altFaceHom a m 2 2 := by
change (Rep.resFunctor _).map (boundary a m 2)=_
rw [boundary_three, Functor.map_add, Functor.map_sub]
rw [he, Functor.map_add, Functor.map_sub,
altFace_H0_eq a m 2 (by omega) 1 0, altFace_H0_eq a m 2 (by omega) 2 0,
sub_self,zero_add]
exact altFace_H0_isIso a m 2 hm 0
lemma altColumn_H1_isZero (a p n : ℕ) (hn : 32 ≤ n) (hm : 3*p+1 < p+n) :
IsZero (groupHomology (altColumn a (p+n) p) 1) := by
have := alternatingStabilizer_perfect a p n hn
exact (TrivialHomology.H1_isZero_of_perfect
(G:=MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n))).of_iso
(alternatingOrbitHomologyIso a p n hm 1)
lemma altBoundary_zero_H2_epi (a m : ℕ) (hm : 34 ≤ m) :
Epi ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 2).map (altBoundary a m 0)) := by
have := altBoundary_zero_epi a m (by omega)
have : IsIso (BoundedHomologyEdge.Hmap 0 (altBoundary a m 2)) :=
altBoundary_two_H0_isIso a m (by omega)
have hz : IsZero (groupHomology (altColumn a m 2) 1) := by
have he : m=2+(m-2) := by omega
rw [he]
exact altColumn_H1_isZero a 2 (m-2) (by omega) (by omega)
exact BoundedHomologyEdge.chain_H2_epi (altBoundary a m 0) (altBoundary a m 1)
(altBoundary a m 2) (altBoundary_square a m 0) (altBoundary_square a m 1)
(altBoundary_exact a m 0 (by omega) (by omega))
(altBoundary_exact a m 1 (by omega) (by omega)) hz
end PolygonPlacement.Configuration
end SimpleAmenable
end
end
end OAI
|