{-# OPTIONS --safe --lossy-unification #-}
module Cubical.ZCohomology.CohomologyRings.KleinBottle where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Transport
open import Cubical.Foundations.HLevels
open import Cubical.Relation.Nullary
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Unit
open import Cubical.Data.Bool
open import Cubical.Data.Nat using (ℕ ; zero ; suc ; discreteℕ ; suc-predℕ ; +-comm)
open import Cubical.Data.Int
open import Cubical.Data.Int.IsEven
open import Cubical.Data.Sigma
open import Cubical.Data.Sum
open import Cubical.Data.Vec
open import Cubical.Data.FinData
open import Cubical.Algebra.Group
open import Cubical.Algebra.Group.Morphisms
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Algebra.AbGroup
open import Cubical.Algebra.Group.Instances.Unit
open import Cubical.Algebra.Group.Instances.Bool
open import Cubical.Algebra.Group.Instances.Int
open import Cubical.Algebra.DirectSum.DirectSumHIT.Base
open import Cubical.Algebra.Ring
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.CommRing.FGIdeal
open import Cubical.Algebra.CommRing.Quotient
open import Cubical.Algebra.CommRing.Instances.Int renaming (ℤCommRing to ℤCR)
open import Cubical.Algebra.CommRing.Instances.Polynomials.MultivariatePoly
open import Cubical.Algebra.CommRing.Instances.Polynomials.MultivariatePoly-Quotient
open import Cubical.Algebra.CommRing.Instances.Polynomials.MultivariatePoly-notationZ
open import Cubical.HITs.Truncation
open import Cubical.HITs.SetTruncation as ST
open import Cubical.HITs.PropositionalTruncation as PT
open import Cubical.HITs.SetQuotients as SQ renaming (_/_ to _/sq_)
open import Cubical.HITs.KleinBottle
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.GroupStructure
open import Cubical.ZCohomology.RingStructure.CupProduct
open import Cubical.ZCohomology.RingStructure.RingLaws
open import Cubical.ZCohomology.RingStructure.CohomologyRing
open import Cubical.ZCohomology.Groups.KleinBottle
open import Cubical.ZCohomology.CohomologyRings.CupProductProperties
open Iso
module Equiv-𝕂²-Properties
(e₁ : GroupIso ℤGroup (coHomGr 1 KleinBottle))
(e₂ : GroupIso BoolGroup (coHomGr 2 KleinBottle))
where
private
ℤAG = Ring→AbGroup (CommRing→Ring ℤCR)
<2Y,Y²,XY,X²> : FinVec ℤ[x,y] 4
<2Y,Y²,XY,X²> zero = base (0 ∷ 1 ∷ []) 2
<2Y,Y²,XY,X²> one = base (0 ∷ 2 ∷ []) 1
<2Y,Y²,XY,X²> two = base (1 ∷ 1 ∷ []) 1
<2Y,Y²,XY,X²> three = base (2 ∷ 0 ∷ []) 1
ℤ[X,Y]/<2Y,Y²,XY,X²> : CommRing ℓ-zero
ℤ[X,Y]/<2Y,Y²,XY,X²> = PolyCommRing-Quotient ℤCR <2Y,Y²,XY,X²>
ℤ[x,y]/<2y,y²,xy,x²> : Type ℓ-zero
ℤ[x,y]/<2y,y²,xy,x²> = fst ℤ[X,Y]/<2Y,Y²,XY,X²>
open IsGroupHom
open gradedRingProperties
open GroupStr (snd BoolGroup) using ()
renaming
( 1g to 0gBool
; _·_ to _+Bool_
; inv to -Bool_
; ·Assoc to +BoolAssoc
; ·IdL to +BoolIdL
; ·IdR to +BoolIdR
; is-set to isSetBool )
open CommRingStr (snd ℤCR) using ()
renaming
( 0r to 0ℤ
; 1r to 1ℤ
; _+_ to _+ℤ_
; -_ to -ℤ_
; _·_ to _·ℤ_
; +Assoc to +ℤAssoc
; +IdL to +ℤIdL
; +IdR to +ℤIdR
; +Comm to +ℤComm
; ·Assoc to ·ℤAssoc
; ·IdL to ·ℤIdL
; ·IdR to ·ℤIdR
; ·DistR+ to ·ℤDistR+
; ·Comm to ·ℤComm
; is-set to isSetℤ )
open RingStr (snd (H*R KleinBottle)) using ()
renaming
( 0r to 0H*
; 1r to 1H*
; _+_ to _+H*_
; -_ to -H*_
; _·_ to _cup_
; +Assoc to +H*Assoc
; +IdL to +H*IdL
; +IdR to +H*IdR
; +Comm to +H*Comm
; ·Assoc to ·H*Assoc
; ·IdL to ·H*IdL
; ·IdR to ·H*IdR
; ·DistR+ to ·H*DistR+
; is-set to isSetH* )
open CommRingStr (snd ℤ[X,Y]) using ()
renaming
( 0r to 0Pℤ
; 1r to 1Pℤ
; _+_ to _+Pℤ_
; -_ to -Pℤ_
; _·_ to _·Pℤ_
; +Assoc to +PℤAssoc
; +IdL to +PℤIdL
; +IdR to +PℤIdR
; +Comm to +PℤComm
; ·Assoc to ·PℤAssoc
; ·IdL to ·PℤIdL
; ·IdR to ·PℤIdR
; ·Comm to ·PℤComm
; ·DistR+ to ·PℤDistR+
; is-set to isSetPℤ )
open CommRingStr (snd ℤ[X,Y]/<2Y,Y²,XY,X²>) using ()
renaming
( 0r to 0PℤI
; 1r to 1PℤI
; _+_ to _+PℤI_
; -_ to -PℤI_
; _·_ to _·PℤI_
; +Assoc to +PℤIAssoc
; +IdL to +PℤIIdL
; +IdR to +PℤIIdR
; +Comm to +PℤIComm
; ·Assoc to ·PℤIAssoc
; ·IdL to ·PℤIIdL
; ·IdR to ·PℤIIdR
; ·DistR+ to ·PℤIDistR+
; is-set to isSetPℤI )
e₀ = invGroupIso H⁰-𝕂²≅ℤ
ϕ₀ = fun (fst e₀)
ϕ₀str = snd e₀
ϕ₀⁻¹ = inv (fst e₀)
ϕ₀⁻¹str = snd (invGroupIso e₀)
ϕ₀-sect = rightInv (fst e₀)
ϕ₀-retr = leftInv (fst e₀)
ϕ₁ = fun (fst e₁)
ϕ₁str = snd e₁
ϕ₁⁻¹ = inv (fst e₁)
ϕ₁⁻¹str = snd (invGroupIso e₁)
ϕ₁-sect = rightInv (fst e₁)
ϕ₁-retr = leftInv (fst e₁)
ϕ₂ = fun (fst e₂)
ϕ₂str = snd e₂
ϕ₂⁻¹ = inv (fst e₂)
ϕ₂⁻¹str = snd (invGroupIso e₂)
ϕ₂-sect = rightInv (fst e₂)
ϕ₂-retr = leftInv (fst e₂)
module PblComp
(null-H¹ : (a b : ℤ) → (ϕ₁ a) ⌣ (ϕ₁ b) ≡ 0ₕ 2)
where
ψ₂ : ℤ → Bool
ψ₂ = isEven
ϕ₂∘ψ₂str : IsGroupHom (snd ℤGroup) (ϕ₂ ∘ ψ₂) (snd (coHomGr 2 KleinBottle))
ϕ₂∘ψ₂str = isGroupHomComp (ψ₂ , isEven-GroupMorphism) (ϕ₂ , ϕ₂str)
ℤ[x,y]→H*-𝕂² : ℤ[x,y] → H* KleinBottle
ℤ[x,y]→H*-𝕂² = DS-Rec-Set.f _ _ _ _ isSetH*
0H*
ϕ
_+H*_
+H*Assoc
+H*IdR
+H*Comm
base-neutral-eq
base-add-eq
where
ϕ : _
ϕ (zero ∷ zero ∷ []) a = base 0 (ϕ₀ a)
ϕ (zero ∷ one ∷ []) a = base 2 ((ϕ₂ ∘ ψ₂) a)
ϕ (zero ∷ suc (suc m) ∷ []) a = 0H*
ϕ (one ∷ zero ∷ []) a = base 1 (ϕ₁ a)
ϕ (one ∷ suc m ∷ []) a = 0H*
ϕ (suc (suc n) ∷ m ∷ []) a = 0H*
base-neutral-eq : _
base-neutral-eq (zero ∷ zero ∷ []) = cong (base 0) (pres1 ϕ₀str) ∙ base-neutral _
base-neutral-eq (zero ∷ one ∷ []) = cong (base 2) (pres1 ϕ₂∘ψ₂str) ∙ base-neutral _
base-neutral-eq (zero ∷ suc (suc m) ∷ []) = refl
base-neutral-eq (one ∷ zero ∷ []) = cong (base 1) (pres1 ϕ₁str) ∙ base-neutral _
base-neutral-eq (one ∷ suc m ∷ []) = refl
base-neutral-eq (suc (suc n) ∷ m ∷ []) = refl
base-add-eq : _
base-add-eq (zero ∷ zero ∷ []) a b = base-add _ _ _ ∙ cong (base 0) (sym (pres· ϕ₀str _ _))
base-add-eq (zero ∷ one ∷ []) a b = base-add _ _ _ ∙ cong (base 2) (sym (pres· ϕ₂∘ψ₂str _ _))
base-add-eq (zero ∷ suc (suc m) ∷ []) a b = +H*IdR _
base-add-eq (one ∷ zero ∷ []) a b = base-add _ _ _ ∙ cong (base 1) (sym (pres· ϕ₁str _ _))
base-add-eq (one ∷ suc m ∷ []) a b = +H*IdR _
base-add-eq (suc (suc n) ∷ m ∷ []) a b = +H*IdR _
ℤ[x,y]→H*-𝕂²-pres1 : ℤ[x,y]→H*-𝕂² (1Pℤ) ≡ 1H*
ℤ[x,y]→H*-𝕂²-pres1 = refl
ℤ[x,y]→H*-𝕂²-pres+ : (x y : ℤ[x,y]) → ℤ[x,y]→H*-𝕂² (x +Pℤ y) ≡ ℤ[x,y]→H*-𝕂² x +H* ℤ[x,y]→H*-𝕂² y
ℤ[x,y]→H*-𝕂²-pres+ x y = refl
open pres⌣
ϕ₀-gen : (n : ℕ) → (f : coHom n KleinBottle) → ϕ₀ (pos 1) ⌣ f ≡ f
ϕ₀-gen n = ST.elim (λ _ → isProp→isSet (GroupStr.is-set (snd (coHomGr n KleinBottle)) _ _))
(λ f → cong ∣_∣₂ (funExt (λ x → rUnitₖ n (f x))))
pres·-int : (n m : ℕ) → (a : ℤ) → (k l : ℕ) → (b : ℤ) →
ℤ[x,y]→H*-𝕂² (base (n ∷ m ∷ []) a ·Pℤ base (k ∷ l ∷ []) b)
≡ ℤ[x,y]→H*-𝕂² (base (n ∷ m ∷ []) a) cup ℤ[x,y]→H*-𝕂² (base (k ∷ l ∷ []) b)
pres·-int zero zero a zero zero b = cong (base 0) (ϕₙ⌣ϕₘ _ ϕ₀str _ ϕ₀str _ ϕ₀str (ϕ₀-gen _ _) _ _)
pres·-int zero zero a zero one b = cong (base 2) (ϕₙ⌣ϕₘ _ ϕ₀str _ ϕ₂∘ψ₂str _ ϕ₂∘ψ₂str (ϕ₀-gen _ _) _ _)
pres·-int zero zero a zero (suc (suc l)) b = refl
pres·-int zero zero a one zero b = cong (base 1) (ϕₙ⌣ϕₘ _ ϕ₀str _ ϕ₁str _ ϕ₁str (ϕ₀-gen _ _) _ _)
pres·-int zero zero a one (suc l) b = refl
pres·-int zero zero a (suc (suc k)) l b = refl
pres·-int zero one a zero zero b = cong ℤ[x,y]→H*-𝕂² (·PℤComm (base (0 ∷ 1 ∷ []) a) (base (0 ∷ 0 ∷ []) b))
∙ pres·-int 0 0 b 0 1 a
∙ gradCommRing KleinBottle _ _ _ _
pres·-int zero one a zero one b = sym (base-neutral 4)
∙ cong (base 4) (unitGroupEq (Hⁿ⁺³-𝕂²≅0 1) _ _)
pres·-int zero one a zero (suc (suc l)) b = refl
pres·-int zero one a one zero b = sym (base-neutral 3)
∙ cong (base 3) (unitGroupEq (Hⁿ⁺³-𝕂²≅0 0) _ _)
pres·-int zero one a one (suc l) b = refl
pres·-int zero one a (suc (suc k)) l b = refl
pres·-int zero (suc (suc m)) a zero l b = refl
pres·-int zero (suc (suc m)) a one l b = refl
pres·-int zero (suc (suc m)) a (suc (suc k)) l b = refl
pres·-int one zero a zero zero b = cong ℤ[x,y]→H*-𝕂² (·PℤComm (base (1 ∷ 0 ∷ []) a) (base (0 ∷ 0 ∷ []) b))
∙ pres·-int 0 0 b 1 0 a
∙ gradCommRing KleinBottle _ _ _ _
pres·-int one zero a zero one b = cong ℤ[x,y]→H*-𝕂² (·PℤComm (base (1 ∷ 0 ∷ []) a) (base (0 ∷ 1 ∷ []) b))
∙ pres·-int 0 1 b 1 0 a
∙ gradCommRing KleinBottle _ _ _ _
pres·-int one zero a zero (suc (suc l)) b = refl
pres·-int one zero a one zero b = sym (base-neutral 2)
∙ cong (base 2) (sym (null-H¹ _ _))
pres·-int one zero a one (suc l) b = refl
pres·-int one zero a (suc (suc k)) l b = refl
pres·-int one (suc m) a zero l b = refl
pres·-int one (suc m) a (suc k) l b = refl
pres·-int (suc (suc n)) m a k l b = refl
pres·-base-case-vec : (v : Vec ℕ 2) → (a : ℤ) → (v' : Vec ℕ 2) → (b : ℤ) →
ℤ[x,y]→H*-𝕂² (base v a ·Pℤ base v' b)
≡ ℤ[x,y]→H*-𝕂² (base v a) cup ℤ[x,y]→H*-𝕂² (base v' b)
pres·-base-case-vec (n ∷ m ∷ []) a (k ∷ l ∷ []) b = pres·-int n m a k l b
ℤ[x,y]→H*-𝕂²-pres· : (x y : ℤ[x,y]) → ℤ[x,y]→H*-𝕂² (x ·Pℤ y) ≡ ℤ[x,y]→H*-𝕂² x cup ℤ[x,y]→H*-𝕂² y
ℤ[x,y]→H*-𝕂²-pres· = DS-Ind-Prop.f _ _ _ _
(λ x p q i y j → isSetH* _ _ (p y) (q y) i j)
(λ y → refl)
base-case
λ {U V} ind-U ind-V y → cong₂ _+H*_ (ind-U y) (ind-V y)
where
base-case : _
base-case v a = DS-Ind-Prop.f _ _ _ _ (λ _ → isSetH* _ _)
(sym (RingTheory.0RightAnnihilates (H*R KleinBottle) _))
(λ v' b → pres·-base-case-vec v a v' b )
λ {U V} ind-U ind-V → (cong₂ _+H*_ ind-U ind-V) ∙ sym (·H*DistR+ _ _ _)
ℤ[x,y]→H*-𝕂²-cancel : (x : Fin 4) → ℤ[x,y]→H*-𝕂² (<2Y,Y²,XY,X²> x) ≡ 0H*
ℤ[x,y]→H*-𝕂²-cancel zero = cong (base 2) (pres1 ϕ₂str) ∙ base-neutral _
ℤ[x,y]→H*-𝕂²-cancel one = refl
ℤ[x,y]→H*-𝕂²-cancel two = refl
ℤ[x,y]→H*-𝕂²-cancel three = refl
ℤ[X,Y]→H*-𝕂² : RingHom (CommRing→Ring ℤ[X,Y]) (H*R KleinBottle)
fst ℤ[X,Y]→H*-𝕂² = ℤ[x,y]→H*-𝕂²
snd ℤ[X,Y]→H*-𝕂² = makeIsRingHom ℤ[x,y]→H*-𝕂²-pres1
ℤ[x,y]→H*-𝕂²-pres+
ℤ[x,y]→H*-𝕂²-pres·
ℤ[X,Y]/<2Y,Y²,XY,X²>→H*R-𝕂² : RingHom (CommRing→Ring ℤ[X,Y]/<2Y,Y²,XY,X²>) (H*R KleinBottle)
ℤ[X,Y]/<2Y,Y²,XY,X²>→H*R-𝕂² = Quotient-FGideal-CommRing-Ring.inducedHom
ℤ[X,Y] (H*R KleinBottle) ℤ[X,Y]→H*-𝕂²
<2Y,Y²,XY,X²> ℤ[x,y]→H*-𝕂²-cancel
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² : ℤ[x,y]/<2y,y²,xy,x²> → H* KleinBottle
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² = fst ℤ[X,Y]/<2Y,Y²,XY,X²>→H*R-𝕂²
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²-pres0 : ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² 0PℤI ≡ 0H*
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²-pres0 = refl
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²-pres+ : (x y : ℤ[x,y]/<2y,y²,xy,x²>) →
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² ( x +PℤI y)
≡ ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² x +H* ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² y
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²-pres+ x y = IsRingHom.pres+ (snd ℤ[X,Y]/<2Y,Y²,XY,X²>→H*R-𝕂²) x y
ψ₂⁻¹ : Bool → ℤ
ψ₂⁻¹ false = 1
ψ₂⁻¹ true = 0
private
Λ : (x : Bool) → ℤ[x,y]/<2y,y²,xy,x²>
Λ x = [ (base (0 ∷ 1 ∷ []) (ψ₂⁻¹ x)) ]
Λ-pres+ : (x y : Bool) → Λ x +PℤI Λ y ≡ Λ (x +Bool y)
Λ-pres+ false false = cong [_] (base-add _ _ _)
∙ eq/ (base (0 ∷ 1 ∷ []) 2)
(base (0 ∷ 1 ∷ []) 0)
∣ ((λ {zero → base (0 ∷ 0 ∷ []) 1 ; one → 0Pℤ ; two → 0Pℤ ; three → 0Pℤ}) , helper) ∣₁
where
helper : _
helper = base-add _ _ _
∙ sym (cong₂ _+Pℤ_ refl (+PℤIdL _ ∙ +PℤIdL _ ∙ +PℤIdL _) ∙ +PℤIdR _)
Λ-pres+ false true = cong [_] (base-add _ _ _)
Λ-pres+ true false = cong [_] (base-add _ _ _)
Λ-pres+ true true = cong [_] (base-add _ _ _)
ϕ⁻¹ : (k : ℕ) → (a : coHom k KleinBottle) → ℤ[x,y]/<2y,y²,xy,x²>
ϕ⁻¹ zero a = [ base (0 ∷ 0 ∷ []) (ϕ₀⁻¹ a) ]
ϕ⁻¹ one a = [ base (1 ∷ 0 ∷ []) (ϕ₁⁻¹ a) ]
ϕ⁻¹ two a = [ base (0 ∷ 1 ∷ []) ((ψ₂⁻¹ ∘ ϕ₂⁻¹) a) ]
ϕ⁻¹ (suc (suc (suc k))) a = 0PℤI
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> : H* KleinBottle → ℤ[x,y]/<2y,y²,xy,x²>
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> = DS-Rec-Set.f _ _ _ _ isSetPℤI
0PℤI
ϕ⁻¹
_+PℤI_
+PℤIAssoc
+PℤIIdR
+PℤIComm
base-neutral-eq
base-add-eq
where
base-neutral-eq : _
base-neutral-eq zero = cong [_] (cong (base {AGP = λ _ → snd ℤAG} (0 ∷ 0 ∷ [])) (pres1 ϕ₀⁻¹str)
∙ (base-neutral _))
base-neutral-eq one = cong [_] (cong (base {AGP = λ _ → snd ℤAG} (1 ∷ 0 ∷ [])) (pres1 ϕ₁⁻¹str)
∙ (base-neutral _))
base-neutral-eq two = cong [_] (cong (base (0 ∷ 1 ∷ [])) (cong ψ₂⁻¹ (pres1 ϕ₂⁻¹str))
∙ base-neutral _)
base-neutral-eq (suc (suc (suc k))) = refl
base-add-eq : _
base-add-eq zero a b = cong [_] (base-add _ _ _ ∙ cong (base (0 ∷ 0 ∷ [])) (sym (pres· ϕ₀⁻¹str _ _)))
base-add-eq one a b = cong [_] (base-add _ _ _ ∙ cong (base (1 ∷ 0 ∷ [])) (sym (pres· ϕ₁⁻¹str _ _)))
base-add-eq two a b = Λ-pres+ (ϕ₂⁻¹ a) (ϕ₂⁻¹ b)
∙ cong [_] (cong (base (0 ∷ 1 ∷ [])) (cong ψ₂⁻¹ (sym (pres· ϕ₂⁻¹str _ _))))
base-add-eq (suc (suc (suc k))) a b = +PℤIIdR _
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²>-pres0 : H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> 0H* ≡ 0PℤI
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²>-pres0 = refl
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²>-pres+ : (x y : H* KleinBottle) →
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> (x +H* y)
≡ (H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> x) +PℤI (H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> y)
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²>-pres+ x y = refl
ψ₂-sect : (x : Bool) → ψ₂ (ψ₂⁻¹ x) ≡ x
ψ₂-sect false = refl
ψ₂-sect true = refl
e-sect-base : (k : ℕ) → (a : coHom k KleinBottle) →
ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² (ϕ⁻¹ k a) ≡ base k a
e-sect-base zero a = cong (base 0) (ϕ₀-sect a)
e-sect-base one a = cong (base 1) (ϕ₁-sect a)
e-sect-base two a = cong (base 2) (cong ϕ₂ (ψ₂-sect _) ∙ ϕ₂-sect a)
e-sect-base (suc (suc (suc k))) a = sym (base-neutral (suc (suc (suc k))))
∙ cong (base (suc (suc (suc k)))) (unitGroupEq (Hⁿ⁺³-𝕂²≅0 k) _ _)
e-sect : (x : H* KleinBottle) → ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² (H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> x) ≡ x
e-sect = DS-Ind-Prop.f _ _ _ _ (λ _ → isSetH* _ _)
refl
e-sect-base
λ {U V} ind-U ind-V → ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²-pres+ _ _ ∙ cong₂ _+H*_ ind-U ind-V
e-retr-ψ₂-false : (a : ℤ) → (isEven a ≡ false) → Λ (ψ₂ a) ≡ [ base (0 ∷ 1 ∷ []) a ]
e-retr-ψ₂-false a x = cong [_] (cong (base (0 ∷ 1 ∷ [])) (cong ψ₂⁻¹ x))
∙ eq/ (base (0 ∷ 1 ∷ []) 1) (base (0 ∷ 1 ∷ []) a)
∣ ((λ {zero → base (0 ∷ 0 ∷ []) (-ℤ m) ; one → 0Pℤ ; two → 0Pℤ ; three → 0Pℤ}) , helper) ∣₁
where
m = fst (isEvenFalse a x)
helper : _
helper = base-add _ _ _
∙ cong (base (0 ∷ 1 ∷ [])) (cong (λ X → 1 +ℤ (-ℤ X)) (snd (isEvenFalse a x))
∙ cong (λ X → 1 +ℤ X) (-Dist+ _ _)
∙ +ℤAssoc _ _ _
∙ +ℤIdL _)
∙ sym (cong₂ _+Pℤ_ (cong (base (0 ∷ 1 ∷ [])) (sym (-DistL· _ _) ∙ cong -ℤ_ (·ℤComm _ _)))
(+PℤIdL _ ∙ +PℤIdL _ ∙ +PℤIdL _)
∙ +PℤIdR _)
e-retr-ψ₂-true : (a : ℤ) → (isEven a ≡ true) → Λ (ψ₂ a) ≡ [ base (0 ∷ 1 ∷ []) a ]
e-retr-ψ₂-true a x = cong [_] (cong (base (0 ∷ 1 ∷ [])) (cong ψ₂⁻¹ x))
∙ eq/ (base (0 ∷ 1 ∷ []) 0) (base (0 ∷ 1 ∷ []) a)
∣ ((λ {zero → base (0 ∷ 0 ∷ []) (-ℤ m) ; one → 0Pℤ ; two → 0Pℤ ; three → 0Pℤ}) , helper) ∣₁
where
m = fst (isEvenTrue a x)
helper : _
helper = base-add _ _ _
∙ cong (base (0 ∷ 1 ∷ [])) (+ℤIdL _ ∙ cong -ℤ_ (snd (isEvenTrue a x)))
∙ sym (cong₂ _+Pℤ_ (cong (base (0 ∷ 1 ∷ [])) (sym (-DistL· _ _) ∙ cong -ℤ_ (·ℤComm _ _)))
(+PℤIdL _ ∙ +PℤIdL _ ∙ +PℤIdL _)
∙ +PℤIdR _)
e-retr-ψ₂ : (a : ℤ) → ((isEven a ≡ false) ⊎ (isEven a ≡ true)) → Λ (ψ₂ a) ≡ [ base (0 ∷ 1 ∷ []) a ]
e-retr-ψ₂ a (inl x) = e-retr-ψ₂-false a x
e-retr-ψ₂ a (inr x) = e-retr-ψ₂-true a x
e-retr-base : (v : Vec ℕ 2) → (a : ℤ) →
H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> (ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² [ base v a ]) ≡ [ base v a ]
e-retr-base (zero ∷ zero ∷ []) a = cong [_] (cong (base (0 ∷ 0 ∷ [])) (ϕ₀-retr a))
e-retr-base (zero ∷ one ∷ []) a = cong [_] (cong (base (0 ∷ 1 ∷ [])) (cong ψ₂⁻¹ (ϕ₂-retr (ψ₂ a))))
∙ e-retr-ψ₂ a (dichotomyBoolSym (isEven a))
e-retr-base (zero ∷ suc (suc m) ∷ []) a = eq/ _ _ ∣ (v , helper) ∣₁
where
v = λ { zero → 0Pℤ ; one → base (0 ∷ m ∷ []) (-ℤ a) ; two → 0Pℤ ; three → 0Pℤ }
helper : _
helper = +PℤIdL _ ∙ sym (+PℤIdL _
∙ cong₂ _+Pℤ_ (cong₂ base (cong (λ X → 0 ∷ X ∷ []) (+-comm _ _)) (·ℤIdR _))
(+PℤIdL _ ∙ +PℤIdL _) ∙ +PℤIdR _)
e-retr-base (one ∷ zero ∷ []) a = cong [_] (cong (base (1 ∷ 0 ∷ [])) (ϕ₁-retr a))
e-retr-base (one ∷ suc m ∷ []) a = eq/ _ _ ∣ (v , helper) ∣₁
where
v = λ { zero → 0Pℤ ; one → 0Pℤ ; two → base (0 ∷ m ∷ []) (-ℤ a) ; three → 0Pℤ }
helper : _
helper = +PℤIdL _ ∙ sym (+PℤIdL _ ∙ +PℤIdL _
∙ cong₂ _+Pℤ_ (cong₂ base (cong (λ X → 1 ∷ X ∷ []) (+-comm _ _)) (·ℤIdR _)) (+PℤIdL _) ∙ +PℤIdR _)
e-retr-base (suc (suc n) ∷ m ∷ []) a = eq/ _ _ ∣ (v , helper) ∣₁
where
v = λ {zero → 0Pℤ ; one → 0Pℤ ; two → 0Pℤ ; three → base (n ∷ m ∷ []) (-ℤ a) }
helper : _
helper = +PℤIdL _ ∙ sym (+PℤIdL _ ∙ +PℤIdL _ ∙ +PℤIdL _ ∙ +PℤIdR _
∙ cong₂ base (cong₂ (λ X → λ Y → X ∷ Y ∷ []) (+-comm _ _) (+-comm _ _)) (·ℤIdR _))
e-retr : (x : ℤ[x,y]/<2y,y²,xy,x²>) → H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²> (ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂² x) ≡ x
e-retr = SQ.elimProp (λ _ → isSetPℤI _ _)
(DS-Ind-Prop.f _ _ _ _ (λ _ → isSetPℤI _ _)
refl
e-retr-base
λ {U V} ind-U ind-V → cong₂ _+PℤI_ ind-U ind-V)
module _ where
open Equiv-𝕂²-Properties (invGroupIso H¹-𝕂²≅ℤ) (invGroupIso H²-𝕂²≅Bool)
open pres⌣trivial
open PblComp (λ a b → sym (ϕₙ⌣ϕₘ-0 ϕ₁ ϕ₁str ϕ₁ ϕ₁str trivial-cup a b))
𝕂²-CohomologyRing : RingEquiv (CommRing→Ring ℤ[X,Y]/<2Y,Y²,XY,X²>) (H*R KleinBottle)
fst 𝕂²-CohomologyRing = isoToEquiv is
where
is : Iso ℤ[x,y]/<2y,y²,xy,x²> (H* KleinBottle)
fun is = ℤ[x,y]/<2y,y²,xy,x²>→H*-𝕂²
inv is = H*-𝕂²→ℤ[x,y]/<2y,y²,xy,x²>
rightInv is = e-sect
leftInv is = e-retr
snd 𝕂²-CohomologyRing = snd ℤ[X,Y]/<2Y,Y²,XY,X²>→H*R-𝕂²
CohomologyRing-𝕂² : RingEquiv (H*R KleinBottle) (CommRing→Ring ℤ[X,Y]/<2Y,Y²,XY,X²>)
CohomologyRing-𝕂² = RingEquivs.invRingEquiv 𝕂²-CohomologyRing