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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonFixedTrackProgress
import OAI.GroupTheory.SimpleAmenable.Homology.ConfigurationEquivariance
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ConfigurationResolution
open Classical CategoryTheory Finsupp
namespace ConfigurationChains
variable {V : Type*} {R : V → V → Prop}
noncomputable def complex (R : V → V → Prop) : ChainComplex (ModuleCat ℤ) ℕ :=
ChainComplex.of (fun n => ModuleCat.of ℤ (degree R n))
(fun n => ModuleCat.ofHom (degreeBoundary R n)) (fun n => by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro c
exact degreeBoundary_square R n c)
noncomputable def representation {G : Type*} [Monoid G] [MulAction G V]
(hR : ∀g : G,∀v w,R v w → R (g • v) (g • w)) (n : ℕ) :
Representation ℤ G (degree R n) where
toFun g := degreePush (fun v => g • v) (hR g) n
map_one' := by
apply LinearMap.ext
intro c
apply Subtype.ext
change push (fun v : V => (1:G) • v) c.val=c.val
have he : (fun v : V => (1:G) • v)=id := funext (fun v => one_smul G v)
rw [he]
exact push_id c.val
map_mul' g h := by
apply LinearMap.ext
intro c
apply Subtype.ext
change push (fun v : V => (g*h) • v) c.val=push (fun v => g • v) (push (fun v => h • v) c.val)
rw [push_comp]
simp only [Function.comp_def, mul_smul]
@[simp] theorem representation_boundary {G : Type*} [Monoid G] [MulAction G V]
(hR : ∀g : G,∀v w,R v w → R (g • v) (g • w)) (g : G) (n : ℕ)
(c : degree R (n+1)) :
representation hR n g (degreeBoundary R n c)=
degreeBoundary R n (representation hR (n+1) g c) :=
degreePush_boundary _ (hR g) n c
noncomputable def simplexModuleEquiv (R : V → V → Prop) (n : ℕ) :
degree R n ≃ₗ[ℤ] ({l : List V // l.Pairwise R ∧ l.length=n} →₀ ℤ) := by
have he : degree R n=Finsupp.supported ℤ ℤ {l : List V | l.Pairwise R ∧ l.length=n} := by
ext c
rw [Finsupp.mem_supported]
rfl
exact (LinearEquiv.ofEq _ _ he).trans (Finsupp.supportedEquivFinsupp _)
theorem lowDegree_exact
(hc : ∀S : Finset V,S.card≤31 → ∃v,∀w∈S,R v w)
{n : ℕ} (hn : n≤2) :
Function.Exact (degreeBoundary R (n+1)) (degreeBoundary R n) := by
intro c
constructor
· intro hz
have hz' : boundary c.val=0 := congrArg Subtype.val hz
obtain ⟨d,hd,he⟩ := low_degree_filling hc (by omega : n+1≤3) c.property hz'
exact ⟨⟨d,hd⟩,Subtype.ext he⟩
· rintro ⟨d,rfl⟩
exact degreeBoundary_square R n d
end ConfigurationChains
namespace PolygonPlacement
noncomputable def configurationRepresentation (a m n : ℕ) :
Representation ℤ (polygonFullGroup a m) (ConfigurationChains.degree (@Apart a m) n) :=
ConfigurationChains.representation (fun g v w h => (apart_smul_iff g v w).mpr h) n
end PolygonPlacement
end ConfigurationResolution
end SimpleAmenable
end
end
end OAI
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