File size: 3,554 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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.CentralCovers.PolygonCommutingRepresentatives
import OAI.GroupTheory.SimpleAmenable.Homology.FullGroupStability
namespace OAI
section
section
open Classical CategoryTheory CategoryTheory.Limits Representation Rep Finsupp
namespace SimpleAmenable
namespace PolygonTracks
variable {a m n : ℕ}
noncomputable def castFull (a : ℕ) {m n : ℕ} (h : m=n) :
polygonFullGroup a m ≃* polygonFullGroup a n := h ▸ MulEquiv.refl _
@[simp] lemma castFull_apply (h : m=n) (f : polygonFullGroup a m) (x : TrackPoint a m) :
(castFull a h f).val (Fin.cast h x.1,x.2)=
(Fin.cast h (f.val x).1,(f.val x).2) := by subst n; rfl
lemma stabilize_add (a k l n : ℕ) : stabilize a (k+l) n =
(castFull a (Nat.add_assoc k l n).symm).toMonoidHom.comp
((stabilize a k (l+n)).comp (stabilize a l n)) := by
let h : k+(l+n)=(k+l)+n := (Nat.add_assoc k l n).symm
apply MonoidHom.ext
intro f
apply Subtype.ext
apply Equiv.ext
intro x
obtain ⟨i,hi⟩ := (finCongr h).surjective x.1
change (stabilize a (k+l) n f).val x =
(castFull a h (stabilize a k (l+n) (stabilize a l n f))).val x
have hx : x=(Fin.cast h i,x.2) := Prod.ext hi.symm rfl
rw [hx, castFull_apply (x:=(i,x.2))]
induction i using Fin.addCases with
| left i =>
have he : Fin.cast h (i.castAdd (l+n))=(i.castAdd l).castAdd n := Fin.ext rfl
rw [he, stabilize_left (x:=((i.castAdd l),x.2)), stabilize_left (x:=(i,x.2))]
exact Prod.ext he.symm rfl
| right i =>
rw [stabilize_right (x:=(i,x.2))]
induction i using Fin.addCases with
| left i =>
have he : Fin.cast h ((i.castAdd n).natAdd k)=(i.natAdd k).castAdd n := Fin.ext rfl
rw [he, stabilize_left (x:=(i.natAdd k,x.2)), stabilize_left (x:=(i,x.2))]
exact Prod.ext he.symm rfl
| right i =>
have he : Fin.cast h ((i.natAdd l).natAdd k)=i.natAdd (k+l) := by
apply Fin.ext
simp only [Fin.val_cast,Fin.val_natAdd]
omega
rw [he, stabilize_right (x:=(i,x.2)), stabilize_right (x:=(i,x.2))]
apply Prod.ext
· apply Fin.ext
simp only [Fin.val_cast,Fin.val_natAdd]
omega
· rfl
lemma stabilize_zero_bijective (a n : ℕ) : Function.Bijective (stabilize a 0 n) := by
refine ⟨stabilize_injective,?_⟩
intro f
let g : bankFixer a 0 n := ⟨f,fun x => Fin.elim0 x.1⟩
exact ⟨restrict g,stabilize_restrict g⟩
lemma stabilize_H1_isIso (a k n : ℕ) (hn : 32 ≤ n) :
IsIso (TrivialHomology.map (stabilize a k n) 1) := by
induction k with
| zero =>
exact TrivialHomology.isIso_equiv
(MulEquiv.ofBijective (stabilize a 0 n) (stabilize_zero_bijective a n)) 1
| succ k ih =>
have he : IsIso (TrivialHomology.map (stabilize a (1+k) n) 1) := by
rw [stabilize_add, TrivialHomology.map_comp, TrivialHomology.map_comp]
have := TrivialHomology.isIso_equiv (castFull a (Nat.add_assoc 1 k n).symm) 1
have := PolygonPlacement.Configuration.stabilize_H1_isIso a (k+n) (by omega)
have := ih
infer_instance
rw [show k+1=1+k by omega]
exact he
lemma stabilize_alternating_comap (a k n : ℕ) (hn : 32 ≤ n) :
(polygonAlternatingGroup a (k+n)).comap (stabilize a k n)=polygonAlternatingGroup a n := by
have := stabilize_H1_isIso a k n hn
rw [← polygonFullGroup_commutator_eq a n (by omega),
← polygonFullGroup_commutator_eq a (k+n) (by omega)]
exact TrivialHomology.commutator_comap_of_mono (stabilize a k n)
end PolygonTracks
end SimpleAmenable
end
end
end OAI
|