module Cubical.Homotopy.Hopf where
open import Cubical.Homotopy.HSpace
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Function
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Univalence
open import Cubical.Data.Sigma
open import Cubical.Data.Unit
open import Cubical.Data.Int hiding (_·_)
open import Cubical.HITs.Pushout.Flattening
open import Cubical.HITs.Pushout
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp
open import Cubical.HITs.S1
open import Cubical.HITs.S2
open import Cubical.HITs.S3
open import Cubical.HITs.PropositionalTruncation
renaming (rec to pRec ; elim to pElim)
open import Cubical.HITs.Join
open import Cubical.HITs.Interval
renaming ( zero to I0 ; one to I1 )
open Iso
open HSpace
open AssocHSpace
private
retEq≡secEq : ∀ {ℓ} {A B : Type ℓ} (e : A ≃ B)
→ (x : _) → secEq e (e .fst x) ≡ cong (e .fst) (retEq e x)
retEq≡secEq {A = A} =
EquivJ (λ B e → (x : _) → secEq e (e .fst x) ≡ cong (e .fst) (retEq e x))
λ _ → refl
module Hopf {ℓ : Level} {A : Pointed ℓ} {e : HSpace A}
(e-ass : AssocHSpace e) (conA : ((x y : typ A) → ∥ x ≡ y ∥₁)) where
isEquiv-μ : (x : typ A) → isEquiv (λ z → (μ e z x))
isEquiv-μ x = pRec (isPropIsEquiv _)
(J (λ x _ → isEquiv (λ z → μ e z x))
(subst isEquiv (funExt (λ z → sym (μᵣ e z)))
(idIsEquiv (typ A))))
(conA (pt A) x)
isEquiv-μ' : (x : typ A) → isEquiv (μ e x)
isEquiv-μ' x =
pRec (isPropIsEquiv _)
(J (λ x _ → isEquiv (μ e x))
(subst isEquiv (funExt (λ x → sym (μₗ e x))) (idIsEquiv (typ A))))
(conA (pt A) x)
μ-eq : (x : typ A) → typ A ≃ typ A
μ-eq x = (λ z → μ e z x) , (isEquiv-μ x)
μ-eq' : (x : typ A) → typ A ≃ typ A
μ-eq' x = μ e x , isEquiv-μ' x
Hopf : Susp (typ A) → Type ℓ
Hopf north = typ A
Hopf south = typ A
Hopf (merid a i₁) = ua (μ-eq a) i₁
TotalSpaceHopfPush : Type _
TotalSpaceHopfPush =
Pushout {A = typ A × typ A} fst λ x → μ e (fst x) (snd x)
TotalSpaceHopfPush→TotalSpace :
TotalSpaceHopfPush → Σ[ x ∈ Susp (typ A) ] Hopf x
TotalSpaceHopfPush→TotalSpace (inl x) = north , x
TotalSpaceHopfPush→TotalSpace (inr x) = south , x
TotalSpaceHopfPush→TotalSpace (push (x , y) i₁) =
merid y i₁ , ua-gluePt (μ-eq y) i₁ x
joinIso₁ : Iso (TotalSpaceHopfPush) (join (typ A) (typ A))
joinIso₁ = theIso
where
F : TotalSpaceHopfPush → join (typ A) (typ A)
F (inl x) = inl x
F (inr x) = inr x
F (push (a , x) i) = push a (μ e a x) i
G : join (typ A) (typ A) → TotalSpaceHopfPush
G (inl x) = inl x
G (inr x) = inr x
G (push a b i) =
(push (a , invEq (μ-eq' a) b) ∙ cong inr (secEq (μ-eq' a) b)) i
s : section F G
s (inl x) = refl
s (inr x) = refl
s (push a b i) j =
hcomp (λ k → λ { (i = i0) → inl a
; (i = i1) → inr (secEq (μ-eq' a) b (j ∨ k))
; (j = i0) → F (compPath-filler
(push (a , invEq (μ-eq' a) b))
(cong inr (secEq (μ-eq' a) b)) k i)
; (j = i1) → push a b i})
(hcomp (λ k → λ { (i = i0) → inl a
; (i = i1) → inr (secEq (μ-eq' a) b (~ k ∨ j))
; (j = i0) → push a (secEq (μ-eq' a) b (~ k)) i
; (j = i1) → push a b i})
(push a b i))
r : retract F G
r (inl x) = refl
r (inr x) = refl
r (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (μ e x y)
; (j = i0) → (push (x , invEq (μ-eq' x) (μ e x y))
∙ (λ i₁ → inr (retEq≡secEq (μ-eq' x) y (~ k) i₁))) i
; (j = i1) → push (x , y) i})
(hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (μ e x (retEq (μ-eq' x) y k))
; (j = i1) → push (x , retEq (μ-eq' x) y k) i})
((push (x , invEq (μ-eq' x) (μ e x y))) i))
theIso : Iso TotalSpaceHopfPush (join (typ A) (typ A))
fun theIso = F
inv theIso = G
sec theIso = s
ret theIso = r
isEquivTotalSpaceHopfPush→TotalSpace :
isEquiv TotalSpaceHopfPush→TotalSpace
isEquivTotalSpaceHopfPush→TotalSpace =
isoToIsEquiv theIso
where
inv' : _ → _
inv' (north , y) = inl y
inv' (south , y) = inr y
inv' (merid a i , y) =
hcomp (λ k → λ { (i = i0) → push (y , a) (~ k)
; (i = i1) → inr y})
(inr (ua-unglue (μ-eq a) i y))
where
pp : PathP (λ i → ua (μ-eq a) i → TotalSpaceHopfPush)
inl inr
pp = ua→ {e = μ-eq a} {B = λ _ → TotalSpaceHopfPush} λ b → push (b , a)
sect : (x : _) → TotalSpaceHopfPush→TotalSpace (inv' x) ≡ x
sect (north , x) = refl
sect (south , x) = refl
sect (merid a i , y) j =
hcomp (λ k → λ { (i = i0) → merid a (~ k ∧ ~ j)
, ua-gluePt (μ-eq a) (~ k ∧ ~ j) y
; (i = i1) → south , y
; (j = i0) →
TotalSpaceHopfPush→TotalSpace
(hfill (λ k → λ { (i = i0) → push (y , a) (~ k)
; (i = i1) → inr y})
(inS (inr (ua-unglue (μ-eq a) i y)))
k)
; (j = i1) → merid a i , y})
((merid a (i ∨ ~ j)) , lem (μ-eq a) i j y)
where
lem : ∀ {ℓ} {A B : Type ℓ} (e : A ≃ B) →
PathP (λ i → PathP (λ j → (y : ua e i) → ua e (i ∨ ~ j))
(λ y → ua-unglue e i y)
λ y → y)
(λ j y → ua-gluePt e (~ j) y)
refl
lem {A = A} {B = B} =
EquivJ (λ B e → PathP (λ i → PathP (λ j → (y : ua e i) → ua e (i ∨ ~ j))
(λ y → ua-unglue e i y)
λ y → y)
(λ j y → ua-gluePt e (~ j) y)
refl)
λ i j a → ua-gluePt (idEquiv B) (i ∨ ~ j) (ua-unglue (idEquiv B) i a)
retr : retract TotalSpaceHopfPush→TotalSpace inv'
retr (inl x) = refl
retr (inr x) = refl
retr (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → push (x , y) (~ k)
; (i = i1) → inr (μ e x y)
; (j = i1) → push (x , y) (i ∨ ~ k)})
(inr (μ e x y))
theIso : Iso TotalSpaceHopfPush (Σ (Susp (typ A)) Hopf)
fun theIso = TotalSpaceHopfPush→TotalSpace
inv theIso = inv'
sec theIso = sect
ret theIso = retr
IsoTotalSpaceJoin : Iso (Σ[ x ∈ Susp (typ A) ] Hopf x) (join (typ A) (typ A))
IsoTotalSpaceJoin =
compIso (equivToIso (invEquiv (_ , isEquivTotalSpaceHopfPush→TotalSpace)))
joinIso₁
induced : TotalSpaceHopfPush → Susp (typ A)
induced = fst ∘ TotalSpaceHopfPush→TotalSpace
ua-lem : (x y z : typ A) → (i j : I) → ua (μ-eq y) i
ua-lem x y z i j =
fill (λ k → ua (μ-eq y) i)
(λ j → λ { (i = i0) → μ e z x
; (i = i1) → μ-assoc e-ass z x y j})
(inS (ua-gluePt (μ-eq y) i (μ e z x)))
j
TotalSpaceHopfPush→≃Hopf : (x : TotalSpaceHopfPush) → typ A ≃ Hopf (induced x)
TotalSpaceHopfPush→≃Hopf (inl x) = μ-eq x
TotalSpaceHopfPush→≃Hopf (inr x) = μ-eq x
TotalSpaceHopfPush→≃Hopf (push (x , y) i₁) = pp x y i₁
where
pp : (x y : _) → PathP (λ i → typ A ≃ ua (μ-eq y) i) (μ-eq x) (μ-eq (μ e x y))
pp x y = ΣPathP (P , help)
where
P : PathP (λ z → typ A → ua (μ-eq y) z) (fst (μ-eq x))
(fst (μ-eq (μ e x y)))
P i z = ua-lem x y z i i1
abstract
help : PathP (λ i₂ → isEquiv (P i₂)) (snd (μ-eq x))
(snd (μ-eq (μ e x y)))
help = toPathP (isPropIsEquiv _ _ _)
Push→TotalSpaceHopf : (a : typ A) (x : TotalSpaceHopfPush)
→ Σ[ x ∈ Susp (typ A) ] Hopf x
Push→TotalSpaceHopf a x = (induced x) , fst (TotalSpaceHopfPush→≃Hopf x) a
Push→TotalSpaceHopf-equiv : (a : typ A) → isEquiv (Push→TotalSpaceHopf a)
Push→TotalSpaceHopf-equiv a = pRec (isPropIsEquiv _)
(J (λ a _ → isEquiv (Push→TotalSpaceHopf a))
(subst isEquiv (sym main)
isEquivTotalSpaceHopfPush→TotalSpace))
(conA (pt A) a)
where
lem₁ : (x : _) → fst ((Push→TotalSpaceHopf (pt A)) x)
≡ fst (TotalSpaceHopfPush→TotalSpace x)
lem₁ (inl x) = refl
lem₁ (inr x) = refl
lem₁ (push a i) = refl
lem₂ : (x : _)
→ PathP (λ i → Hopf (lem₁ x i))
(snd ((Push→TotalSpaceHopf (pt A)) x))
(snd (TotalSpaceHopfPush→TotalSpace x))
lem₂ (inl x) = μₗ e x
lem₂ (inr x) = μₗ e x
lem₂ (push (x , y) i) j =
hcomp (λ k → λ {(i = i0) → μₗ e x j
; (i = i1) → μ-assoc-filler e-ass x y j k
; (j = i0) → ua-lem x y (pt A) i k
; (j = i1) → ua-gluePt (μ-eq y) i x})
(ua-gluePt (μ-eq y) i (μₗ e x j))
main : Push→TotalSpaceHopf (pt A) ≡ TotalSpaceHopfPush→TotalSpace
main i x = (lem₁ x i) , (lem₂ x i)
TotalSpaceHopfPush² : Type _
TotalSpaceHopfPush² = Pushout {A = TotalSpaceHopfPush} (λ _ → tt) induced
P : TotalSpaceHopfPush² → Type _
P (inl x) = typ A
P (inr x) = Hopf x
P (push a i) = ua (TotalSpaceHopfPush→≃Hopf a) i
TotalSpacePush² : Type _
TotalSpacePush² = Σ[ x ∈ TotalSpaceHopfPush² ] P x
TotalSpacePush²' : Type _
TotalSpacePush²' =
Pushout {A = typ A × TotalSpaceHopfPush}
{C = Σ[ x ∈ Susp (typ A) ] Hopf x}
fst
λ x → Push→TotalSpaceHopf (fst x) (snd x)
IsoTotalSpacePush²TotalSpacePush²' : Iso TotalSpacePush² TotalSpacePush²'
IsoTotalSpacePush²TotalSpacePush²' =
compIso iso₂ (compIso (equivToIso fl.flatten) iso₁)
where
module fl =
FlatteningLemma (λ _ → tt) induced (λ x → typ A)
Hopf TotalSpaceHopfPush→≃Hopf
iso₁ : Iso (Pushout fl.Σf fl.Σg) TotalSpacePush²'
fun iso₁ (inl x) = inl (snd x)
fun iso₁ (inr x) = inr x
fun iso₁ (push a i) = push ((snd a) , (fst a)) i
inv iso₁ (inl x) = inl (tt , x)
inv iso₁ (inr x) = inr x
inv iso₁ (push a i) = push (snd a , fst a) i
sec iso₁ (inl x) = refl
sec iso₁ (inr x) = refl
sec iso₁ (push a i) = refl
ret iso₁ (inl x) = refl
ret iso₁ (inr x) = refl
ret iso₁ (push a i) = refl
iso₂ : Iso TotalSpacePush² (Σ (Pushout (λ _ → tt) induced) fl.E)
fun iso₂ (inl x , y) = inl x , y
fun iso₂ (inr x , y) = inr x , y
fun iso₂ (push a i , y) = push a i , y
inv iso₂ (inl x , y) = inl x , y
inv iso₂ (inr x , y) = inr x , y
inv iso₂ (push a i , y) = push a i , y
sec iso₂ (inl x , snd₁) = refl
sec iso₂ (inr x , snd₁) = refl
sec iso₂ (push a i , snd₁) = refl
ret iso₂ (inl x , snd₁) = refl
ret iso₂ (inr x , snd₁) = refl
ret iso₂ (push a i , snd₁) = refl
F : TotalSpacePush²'
→ (Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
F (inl x) = inl x
F (inr x) = inr x
F (push (x , y) i) = push (x , Push→TotalSpaceHopf x y) i
G : (Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
→ TotalSpacePush²'
G (inl x) = inl x
G (inr x) = inr x
G (push (x , y) i) =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_ , Push→TotalSpaceHopf-equiv x) y k)})
(push (x , invEq (_ , Push→TotalSpaceHopf-equiv x) y) i)
IsoTotalSpacePush²'ΣPush : Iso TotalSpacePush²'
(Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
fun IsoTotalSpacePush²'ΣPush = F
inv IsoTotalSpacePush²'ΣPush = G
sec IsoTotalSpacePush²'ΣPush (inl x) = refl
sec IsoTotalSpacePush²'ΣPush (inr x) = refl
sec IsoTotalSpacePush²'ΣPush (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_ , Push→TotalSpaceHopf-equiv x) y k)
; (j = i0) → F (
hfill (λ k →
λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_
, Push→TotalSpaceHopf-equiv x) y k)})
(inS (push (x
, invEq (_
, Push→TotalSpaceHopf-equiv x) y) i)) k)
; (j = i1)
→ push (x
, (secEq (_
, Push→TotalSpaceHopf-equiv x) y k)) i})
(push (x , (secEq (_ , Push→TotalSpaceHopf-equiv x) y i0)) i)
ret IsoTotalSpacePush²'ΣPush (inl x) = refl
ret IsoTotalSpacePush²'ΣPush (inr x) = refl
ret IsoTotalSpacePush²'ΣPush (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (secEq (Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
(Push→TotalSpaceHopf x y) (j ∨ k))
; (j = i1) → push (x , y) i})
(hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (retEq≡secEq
(Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
y (~ k) j)
; (j = i1) → push (x , y) i
; (j = i0) → push (x , invEq
(Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
(Push→TotalSpaceHopf x y)) i})
(push (x , retEq (Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x) y j) i))
joinIso₂ : Iso TotalSpacePush² (join (typ A) (join (typ A) (typ A)))
joinIso₂ =
compIso IsoTotalSpacePush²TotalSpacePush²'
(compIso IsoTotalSpacePush²'ΣPush
(compIso (equivToIso (joinPushout≃join _ _))
(pathToIso (cong (join (typ A))
(isoToPath IsoTotalSpaceJoin)))))
module S¹Hopf where
Border : (x : S¹) → (j : I) → Partial (j ∨ ~ j) (Σ Type₀ (λ T → T ≃ S¹))
Border x j (j = i0) = S¹ , (x ·_) , rotIsEquiv x
Border x j (j = i1) = S¹ , idEquiv S¹
HopfSuspS¹ : SuspS¹ → Type₀
HopfSuspS¹ north = S¹
HopfSuspS¹ south = S¹
HopfSuspS¹ (merid x j) = Glue S¹ (Border x j)
HopfS² : S² → Type₀
HopfS² base = S¹
HopfS² (surf i j) = Glue S¹ (λ { (i = i0) → _ , idEquiv S¹
; (i = i1) → _ , idEquiv S¹
; (j = i0) → _ , idEquiv S¹
; (j = i1) → _ , _ , rotIsEquiv (loop i) } )
HopfS²' : S² → Type₀
HopfS²' base = S¹
HopfS²' (surf i j) = Glue S¹ (λ { (i = i0) → _ , rotLoopEquiv i0
; (i = i1) → _ , rotLoopEquiv i0
; (j = i0) → _ , rotLoopEquiv i0
; (j = i1) → _ , rotLoopEquiv i } )
TotalHopf : Type₀
TotalHopf = Σ SuspS¹ HopfSuspS¹
filler-1 : I → (j : I) → (y : S¹) → Glue S¹ (Border y j) → join S¹ S¹
filler-1 i j y x = hfill (λ t → λ { (j = i0) → inl (rotInv-1 x y t)
; (j = i1) → inr x })
(inS (push ((unglue (j ∨ ~ j) x) · invLooper y) (unglue (j ∨ ~ j) x) j)) i
TotalHopf→JoinS¹S¹ : TotalHopf → join S¹ S¹
TotalHopf→JoinS¹S¹ (north , x) = inl x
TotalHopf→JoinS¹S¹ (south , x) = inr x
TotalHopf→JoinS¹S¹ (merid y j , x) = filler-1 i1 j y x
JoinS¹S¹→TotalHopf : join S¹ S¹ → TotalHopf
JoinS¹S¹→TotalHopf (inl x) = (north , x)
JoinS¹S¹→TotalHopf (inr x) = (south , x)
JoinS¹S¹→TotalHopf (push y x j) =
(merid (invLooper y · x) j
, glue (λ { (j = i0) → y ; (j = i1) → x }) (rotInv-2 x y j))
fibℤ : S¹ → S¹ → Type₀
fibℤ _ _ = ℤ
S¹→HSet : (A : Type₀) (p : isSet A) (F : S¹ → A) (x : S¹) → F base ≡ F x
S¹→HSet A p F base = refl {x = F base}
S¹→HSet A p F (loop i) = f' i
where
f : PathP (λ i → F base ≡ F (loop i)) refl (cong F loop)
f i = λ j → F (loop (i ∧ j))
L : cong F loop ≡ refl
L = p (F base) (F base) (f i1) refl
f' : PathP (λ i → F base ≡ F (loop i)) (refl {x = F base}) (refl {x = F base})
f' = transport (λ i → PathP (λ j → F base ≡ F (loop j)) refl (L i)) f
constant-loop : (F : S¹ → S¹ → ℤ) → (x y : S¹) → F base base ≡ F x y
constant-loop F x y = L0 ∙ L1
where
p : isSet (S¹ → ℤ)
p = isSetΠ (λ _ → isSetℤ)
L : F base ≡ F x
L = S¹→HSet (S¹ → ℤ) p F x
L0 : F base base ≡ F x base
L0 i = L i base
L1 : F x base ≡ F x y
L1 = S¹→HSet ℤ isSetℤ (F x) y
discretefib : (F : S¹ → S¹ → Type₀) → Type₀
discretefib F = (a : (x y : S¹) → F x y) →
(b : (x y : S¹) → F x y) →
(a base base ≡ b base base) →
(x y : S¹) → a x y ≡ b x y
discretefib-fibℤ : discretefib fibℤ
discretefib-fibℤ a b h x y i =
hcomp (λ t → λ { (i = i0) → constant-loop a x y t
; (i = i1) → constant-loop b x y t })
(h i)
assocFiller-3-aux : I → I → I → I → S¹
assocFiller-3-aux x y j i =
hfill (λ t → λ { (i = i0) → rotInv-1 (loop y) (loop (~ y) · loop x) t
; (i = i1) → rotInv-3 (loop y) (loop x) t
; (x = i0) (y = i0) → base
; (x = i0) (y = i1) → base
; (x = i1) (y = i0) → base
; (x = i1) (y = i1) → base })
(inS ((rotInv-2 (loop x) (loop y) i) · (invLooper (loop (~ y) · loop x)))) j
assocFiller-3-endpoint : (x : S¹) → (y : S¹) → y ≡ y
assocFiller-3-endpoint base base i = base
assocFiller-3-endpoint (loop x) base i = assocFiller-3-aux x i0 i1 i
assocFiller-3-endpoint base (loop y) i = assocFiller-3-aux i0 y i1 i
assocFiller-3-endpoint (loop x) (loop y) i = assocFiller-3-aux x y i1 i
assocFiller-3 : (x : S¹) → (y : S¹) →
PathP (λ j → rotInv-1 y (invLooper y · x) j ≡ rotInv-3 y x j)
(λ i → ((rotInv-2 x y i) · (invLooper (invLooper y · x))))
(assocFiller-3-endpoint x y)
assocFiller-3 base base j i = base
assocFiller-3 (loop x) base j i = assocFiller-3-aux x i0 j i
assocFiller-3 base (loop y) j i = assocFiller-3-aux i0 y j i
assocFiller-3 (loop x) (loop y) j i = assocFiller-3-aux x y j i
assoc-3 : (_ y : S¹) → basedΩS¹ y
assoc-3 x y i = assocFiller-3 x y i1 i
fibℤ≡fibAssoc-3 : fibℤ ≡ (λ _ y → basedΩS¹ y)
fibℤ≡fibAssoc-3 i = λ x y → basedΩS¹≡ℤ y (~ i)
discretefib-fibAssoc-3 : discretefib (λ _ y → basedΩS¹ y)
discretefib-fibAssoc-3 =
transp (λ i → discretefib (fibℤ≡fibAssoc-3 i)) i0 discretefib-fibℤ
assocConst-3 : (x y : S¹) → assoc-3 x y ≡ refl
assocConst-3 x y = discretefib-fibAssoc-3 assoc-3 (λ _ _ → refl) refl x y
assocSquare-3 : I → I → S¹ → S¹ → S¹
assocSquare-3 i j x y = hcomp (λ t → λ { (i = i0) → assocFiller-3 x y j i0
; (i = i1) → assocFiller-3 x y j i1
; (j = i0) → assocFiller-3 x y i0 i
; (j = i1) → assocConst-3 x y t i })
(assocFiller-3 x y j i)
filler-3 : I → I → S¹ → S¹ → join S¹ S¹
filler-3 i j y x =
hcomp (λ t → λ { (i = i0) → filler-1 t j (invLooper y · x)
(glue (λ { (j = i0) → y ; (j = i1) → x })
(rotInv-2 x y j))
; (i = i1) → push (rotInv-3 y x t) x j
; (j = i0) → inl (assocSquare-3 i t x y)
; (j = i1) → inr x })
(push ((rotInv-2 x y (i ∨ j)) · (invLooper (invLooper y · x))) (rotInv-2 x y (i ∨ j)) j)
JoinS¹S¹→TotalHopf→JoinS¹S¹ : ∀ x → TotalHopf→JoinS¹S¹ (JoinS¹S¹→TotalHopf x) ≡ x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (inl x) i = inl x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (inr x) i = inr x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (push y x j) i = filler-3 i j y x
PseudoHopf : Type₀
PseudoHopf = (S¹ × Interval) × S¹
PseudoHopf-π1 : PseudoHopf → S¹
PseudoHopf-π1 ((y , _) , _) = y
PseudoHopf-π2 : PseudoHopf → S¹
PseudoHopf-π2 (_ , x) = x
assocFiller-4-aux : I → I → I → I → S¹
assocFiller-4-aux x y j i =
hfill (λ t → λ { (i = i0) → ((invLooper (loop y · loop x · loop (~ y))) · (loop y · loop x))
· (rotInv-1 (loop x) (loop y) t)
; (i = i1) → (rotInv-4 (loop y) (loop y · loop x) (~ t)) · loop x
; (x = i0) (y = i0) → base
; (x = i0) (y = i1) → base
; (x = i1) (y = i0) → base
; (x = i1) (y = i1) → base })
(inS (rotInv-2 (loop y · loop x) (loop y · loop x · loop (~ y)) i)) j
assocFiller-4-endpoint : (x y : S¹) → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x)
assocFiller-4-endpoint base base i = base
assocFiller-4-endpoint (loop x) base i = assocFiller-4-aux x i0 i1 i
assocFiller-4-endpoint base (loop y) i = assocFiller-4-aux i0 y i1 i
assocFiller-4-endpoint (loop x) (loop y) i = assocFiller-4-aux x y i1 i
assocFiller-4 : (x y : S¹) →
PathP (λ j → ((invLooper (y · x · invLooper y)) · (y · x)) · (rotInv-1 x y j) ≡ (rotInv-4 y (y · x) (~ j)) · x)
(λ i → (rotInv-2 (y · x) (y · x · invLooper y) i))
(assocFiller-4-endpoint x y)
assocFiller-4 base base j i = base
assocFiller-4 (loop x) base j i = assocFiller-4-aux x i0 j i
assocFiller-4 base (loop y) j i = assocFiller-4-aux i0 y j i
assocFiller-4 (loop x) (loop y) j i = assocFiller-4-aux x y j i
assoc-4 : (x y : S¹) → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x)
assoc-4 x y i = assocFiller-4 x y i1 i
fibℤ≡fibAssoc-4 : fibℤ ≡ (λ x y → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x))
fibℤ≡fibAssoc-4 i = λ x y → basedΩS¹≡ℤ (((invLooper (y · x · invLooper y)) · (y · x)) · x) (~ i)
discretefib-fibAssoc-4 : discretefib (λ x y → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x))
discretefib-fibAssoc-4 =
transp (λ i → discretefib (fibℤ≡fibAssoc-4 i)) i0 discretefib-fibℤ
assocConst-4 : (x y : S¹) → assoc-4 x y ≡ refl
assocConst-4 x y = discretefib-fibAssoc-4 assoc-4 (λ _ _ → refl) refl x y
assocSquare-4 : I → I → S¹ → S¹ → S¹
assocSquare-4 i j x y =
hcomp (λ t → λ { (i = i0) → assocFiller-4 x y j i0
; (i = i1) → assocFiller-4 x y j i1
; (j = i0) → assocFiller-4 x y i0 i
; (j = i1) → assocConst-4 x y t i })
(assocFiller-4 x y j i)
filler-4-0 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-0 i j y x =
let x' = unglue (j ∨ ~ j) x in
hfill (λ t → λ { (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0)
, invLooper (y · x · invLooper y) · (y · x) · (rotInv-1 x y t))
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
(inS ((invLooper (x' · invLooper y) · x' , seg j) , rotInv-2 x' (x' · invLooper y) j)) i
filler-4-1 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-1 i j y x =
let x' = unglue (j ∨ ~ j) x in
hfill (λ t → λ { (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0)
, (rotInv-4 y (y · x) (~ t)) · x)
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
(inS ((invLooper (x' · invLooper y) · x' , seg j) , unglue (j ∨ ~ j) x)) i
filler-4-2 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → TotalHopf
filler-4-2 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → JoinS¹S¹→TotalHopf (filler-1 t j y x)
; (i = i1) → (merid (PseudoHopf-π1 (filler-4-0 t j y x)) j
, glue (λ { (j = i0) → rotInv-1 x y t ; (j = i1) → x })
(PseudoHopf-π2 (filler-4-0 t j y x)))
; (j = i0) → (north , rotInv-1 x y t)
; (j = i1) → (south , x) })
(merid (invLooper (x' · invLooper y) · x') j
, glue (λ { (j = i0) → y · x · invLooper y ; (j = i1) → x }) (rotInv-2 x' (x' · invLooper y) j))
filler-4-3 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-3 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → filler-4-0 t j y x
; (i = i1) → filler-4-1 t j y x
; (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0) , assocSquare-4 i t x y)
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
((invLooper (x' · invLooper y) · x' , seg j) , rotInv-2 x' (x' · invLooper y) (i ∨ j))
filler-4-4 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-4 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → filler-4-1 t j y x
; (i = i1) → ((y , seg j) , unglue (j ∨ ~ j) x)
; (j = i0) → ((rotInv-4 y (y · x) i , I0)
, (rotInv-4 y (y · x) (i ∨ ~ t)) · x)
; (j = i1) → ((rotInv-4 y x i , I1) , x) })
((rotInv-4 y x' i , seg j) , x')
filler-4-5 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → TotalHopf
filler-4-5 i j y x =
hcomp (λ t → λ { (i = i0) → filler-4-2 (~ t) j y x
; (i = i1) → (merid (PseudoHopf-π1 (filler-4-4 t j y x)) j
, glue (λ { (j = i0) → x ; (j = i1) → x })
(PseudoHopf-π2 (filler-4-4 t j y x)))
; (j = i0) → (north , x)
; (j = i1) → (south , x) })
(merid (PseudoHopf-π1 (filler-4-3 i j y x)) j
, glue (λ { (j = i0) → x ; (j = i1) → x }) (PseudoHopf-π2 (filler-4-3 i j y x)))
TotalHopf→JoinS¹S¹→TotalHopf : ∀ x → JoinS¹S¹→TotalHopf (TotalHopf→JoinS¹S¹ x) ≡ x
TotalHopf→JoinS¹S¹→TotalHopf (north , x) i = (north , x)
TotalHopf→JoinS¹S¹→TotalHopf (south , x) i = (south , x)
TotalHopf→JoinS¹S¹→TotalHopf (merid y j , x) i = filler-4-5 i j y x
JoinS¹S¹≡TotalHopf : join S¹ S¹ ≡ TotalHopf
JoinS¹S¹≡TotalHopf = isoToPath (iso JoinS¹S¹→TotalHopf
TotalHopf→JoinS¹S¹
TotalHopf→JoinS¹S¹→TotalHopf
JoinS¹S¹→TotalHopf→JoinS¹S¹)
S³≡TotalHopf : S³ ≡ TotalHopf
S³≡TotalHopf = S³≡joinS¹S¹ ∙ JoinS¹S¹≡TotalHopf
open Iso
IsoS³TotalHopf : Iso (S₊ 3) TotalHopf
fun IsoS³TotalHopf x = JoinS¹S¹→TotalHopf (S³→joinS¹S¹ (inv IsoS³S3 x))
inv IsoS³TotalHopf x = fun IsoS³S3 (joinS¹S¹→S³ (TotalHopf→JoinS¹S¹ x))
sec IsoS³TotalHopf x =
cong (JoinS¹S¹→TotalHopf ∘ S³→joinS¹S¹)
(ret IsoS³S3 (joinS¹S¹→S³ (TotalHopf→JoinS¹S¹ x)))
∙∙ cong JoinS¹S¹→TotalHopf
(joinS¹S¹→S³→joinS¹S¹ (TotalHopf→JoinS¹S¹ x))
∙∙ TotalHopf→JoinS¹S¹→TotalHopf x
ret IsoS³TotalHopf x =
cong (fun IsoS³S3 ∘ joinS¹S¹→S³)
(JoinS¹S¹→TotalHopf→JoinS¹S¹ (S³→joinS¹S¹ (inv IsoS³S3 x)))
∙∙ cong (fun IsoS³S3) (S³→joinS¹S¹→S³ (inv IsoS³S3 x))
∙∙ Iso.sec IsoS³S3 x