open import 1Lab.Path.Reasoning
open import 1Lab.Prelude

open import Algebra.Group.Cat.Base
open import Algebra.Group.Concrete
open import Algebra.Group.Homotopy
open import Algebra.Group

open import Data.Set.Truncation

open import Homotopy.Space.Suspension
open import Homotopy.Connectedness
open import Homotopy.Conjugation
open import Homotopy.Truncation
open import Homotopy.Loopspace
open import Homotopy.HSpace
open import Homotopy.Wedge
open import Homotopy.Base
open import Homotopy.Join

open ConcreteGroup

module Homotopy.Space.Suspension.Pi2 {ℓ} (grp : ConcreteGroup ℓ) (hg : HSpace (grp .B)) where

π₂ of a suspension🔗

open ConcreteGroup grp renaming (B to BG ; pt to G₀) using ()
open HSpace {ℓ = ℓ} {A* = BG} hg

private
  G : Type ℓ
  G = ⌞ grp ⌟

  ΣG : Type ℓ
  ΣG = Susp G

  ∥_∥₁ : Type ℓ → Type ℓ
  ∥ X ∥₁ = n-Tr X 3

  μr : ∀ a → ⌞ G ⌟ ≃ ⌞ G ⌟
  μr a = _ , μ-invr a

We will prove that the second homotopy group $π_{2} (Σ G)$ of the suspension of a pointed connected groupoid $G$ with an H-space structure (hence an abelian concrete group) is $Ω G .$

We start by defining a type family Hopf over $Σ G$ which is $G$ on both poles and sends the $x th$ meridian to the H-space multiplication $μ (-, x)$ on the right, which is an equivalence by assumption. We will show, by an encode-decode argument, that the groupoid truncation $∥ north \equiv x ∥_{1}$ is equivalent to the fibre of Hopf over $x .$

As the name implies, the Hopf type family is the synthetic equivalent of the classic Hopf fibration, though here we have evidently generalised it beyond a map $S^{3} \to S^{1} .$ Below we prove that the total space of the Hopf fibration is the join $G * G .$

Hopf : ΣG → Type _
Hopf north       = G
Hopf south       = G
Hopf (merid x i) = ua (μr x) i

To encode, we use truncation recursion, since Hopf is a family of groupoids by construction, and since we have $G_{0} : Hopf (north)$ we can transport it along a $north \equiv x$ to get a point in an arbitrary fibre.

encode' : ∀ x → ∥ north ≡ x ∥₁ → Hopf x
encode' x = n-Tr-rec (tr x) λ p → subst Hopf p G₀ where
  tr : ∀ x → is-groupoid (Hopf x)
  tr = Susp-elim-prop (λ s → hlevel 1) (grp .has-is-groupoid) (grp .has-is-groupoid)

To decode an element of Hopf we use suspension recursion. On the north pole we can use the suspension map $σ : A \to_{*} Ω Σ A;$ on the south pole this is just a meridian; and on the meridians we must prove that these agree. Through a short calculation we can reduce this to a coherence lemma relating the composition of meridians and the H-space multiplication.

decode' : ∀ x → Hopf x → ∥ north ≡ x ∥₁
decode' = Susp-elim _ (n-Tr.inc ∘ suspend BG) (n-Tr.inc ∘ merid) λ x → ua→ λ a → to-pathp $
  inc (subst (north ≡_) (merid x) (suspend BG a)) ≡⟨ ap n-Tr.inc (subst-path-right (suspend BG a) (merid x)) ⟩≡
  inc ((merid a ∙ sym (merid G₀)) ∙ merid x)      ≡˘⟨ ap n-Tr.inc (∙-assoc _ _ _) ⟩≡˘
  inc (merid a ∙ sym (merid G₀) ∙ merid x)        ≡⟨ merid-μ a x ⟩≡
  inc (merid (μ a x))                             ∎
  where

To show this coherence, we can use the wedge connectivity lemma. It will suffice to do so, in turn, when $b = G_{0},$ when $a = G_{0},$ and then to show that these agree. In either case we must show something like $merid a \cdot (merid G_{0})^{- 1} \cdot merid G_{0} = merid μ (a, G_{0}),$ which is easy to do with the pre-existing coherence lemmas ∙∙-introl and ∙∙-intror, and the H-space unit laws.

  merid-μ : ∀ a b → inc (merid a ∙ sym (merid G₀) ∙ merid b) ≡ inc (merid (μ a b))
  merid-μ =
    let
      α = merid G₀

      P : (a b : ⌞ G ⌟) → Type ℓ
      P a b = Path ∥ north ≡ south ∥₁ (inc (merid a ∙ sym α ∙ merid b)) (inc (merid (μ a b)))

      p1 : ∀ a → P a G₀
      p1 a = ap n-Tr.inc $
           sym (double-composite _ _ _)
        ∙∙ sym (∙∙-intror (merid a) (sym α))
        ∙∙ ap merid (sym (idr a))

      p2 : ∀ b → P G₀ b
      p2 b = ap n-Tr.inc $
           sym (double-composite _ _ _)
        ∙∙ sym (∙∙-introl (merid b) α)
        ∙∙ ap merid (sym (idl b))

Moreover, it is easy to show that these agree: by construction we’re left with showing that ∙∙-introl and ∙∙-intror agree when all three paths are the same, and by path induction we may assume that this one path is refl; In that case, they agree definitionally.

      coh : p1 G₀ ≡ p2 G₀
      coh = ap (ap n-Tr.inc) $ ap₂ (_∙∙_∙∙_ (sym (double-composite α (sym α) α)))
        (ap sym (J (λ y p → ∙∙-intror p (sym p) ≡ ∙∙-introl p p) refl α))
        (ap (ap merid ∘ sym) (sym id-coh))

      c = is-connected∙→is-connected (grp .has-is-connected)
    in Wedge.elim {A∙ = BG} {BG} 0 0 c c (λ _ _ → hlevel 2) p1 p2 coh

We have thus constructed maps between $Hopf (x)$ and the truncation of the based path space $∥ north \equiv x ∥_{1} .$ We must then show that these are inverses, which, in both directions, are simple calculations.

opaque
  ΩΣG≃G : ∥ north {A = fst BG} ≡ north ∥₁ ≃ G
  ΩΣG≃G .fst = encode' north

  ΩΣG≃G .snd = is-iso→is-equiv (iso (decode' north) invl (invr north)) where abstract
    invl : ∀ a → encode' north (decode' north a) ≡ a
    invl a = Regularity.fast! (
      Equiv.from (flip μ G₀ , μ-invr G₀) (μ G₀ a) ≡⟨ ap (λ e → Equiv.from e (μ G₀ a)) {x = _ , μ-invr G₀} {y = id≃} (ext idr) ⟩≡
      μ G₀ a                                      ≡⟨ idl a ⟩≡
      a                                           ∎)

To show that decoding inverts encoding, we use the extra generality afforded by the $x$ parameter to apply path induction.

    invr : (x : ΣG) (p : ∥ north ≡ x ∥₁) → decode' x (encode' x p) ≡ p
    invr x = n-Tr-elim! _ $ J
      (λ x p → decode' x (encode' x (inc p)) ≡ inc p)
      (ap n-Tr.inc
        ( ap₂ _∙_ (ap merid (transport-refl _)) refl
        ∙ ∙-invr (merid G₀)))

This equivalence is pointed, almost by definition.

  ΩΣG≃∙G : n-Tr∙ (Ω¹ (Σ¹ BG)) 3 ≃∙ BG
  ΩΣG≃∙G = ΩΣG≃G , transport-refl _

Furthermore, note that, by construction, the inverse pointed map $G \to_{*} ∥ Ω Σ G ∥_{1}$ of this equivalence is none other than suspend∙ followed by the inclusion into the groupoid truncation.

  ΩΣG≃∙G-inv : Equiv∙.from∙ ΩΣG≃∙G ≡ inc∙ ∘∙ suspend∙ BG
  ΩΣG≃∙G-inv = homogeneous-funext∙ λ _ → refl

Finally, we can apply $Ω$ to the equivalence above and use some pre-existing lemmas to show that the set truncation of the double loop space $Ω^{2} Σ G$ is equivalent to $Ω G .$ A couple more short calculations which we omit show that this equivalence preserves path composition, i.e. is an isomorphism of homotopy groups $π_{2} (Σ G) ≅ π_{1} (G),$ and that the inverse map is the expected suspension map $Ω σ$ up to truncation.

Ω²ΣG≃ΩG : ∥ ⌞ Ωⁿ 2 (Σ¹ BG) ⌟ ∥₀ ≃ ⌞ Ω¹ BG ⌟
Ω²ΣG≃ΩG =
  ⌞ πₙ₊₁ 1 (Σ¹ BG) ⌟            ≃⟨ n-Tr-set ⟩≃
  ⌞ n-Tr∙ (Ωⁿ 2 (Σ¹ BG)) 2 ⌟    ≃⟨ n-Tr-Ω¹ (Ω¹ (Σ¹ BG)) 1 .fst ⟩≃
  ⌞ Ω¹ (n-Tr∙ (Ω¹ (Σ¹ BG)) 3) ⌟ ≃⟨ Ω¹-ap ΩΣG≃∙G .fst ⟩≃
  ⌞ Ω¹ BG ⌟                     ≃∎

opaque
  unfolding Ω¹-ap

  Ω²ΣG≃ΩG-pres : is-group-hom
    (πₙ₊₁ 1 (Σ¹ BG) .snd)
    (π₁Groupoid.π₁ BG (grp .has-is-groupoid) .snd)
    (Ω²ΣG≃ΩG .fst)

  Ω²ΣG≃ΩG-inv
    : ∀ (l : ⌞ Ω¹ BG ⌟)
    → Equiv.from Ω²ΣG≃ΩG l ≡ inc (Ω¹-map (suspend∙ BG) .fst l)

  Ω²ΣG≃ΩG-pres = record { pres-⋆ = elim! λ p q → trace p q .snd } where
    open Σ Ω²ΣG≃ΩG renaming (fst to f0) using ()
    instance
      _ : ∀ {n} → H-Level ⌞ G ⌟ (3 + n)
      _ = basic-instance 3 (grp .has-is-groupoid)

    trace : (p q : refl ≡ refl) → (∥ ⌞ Ωⁿ 2 (Σ¹ BG) ⌟ ∥₀ , inc (p ∙ q)) ≃∙ (⌞ Ω¹ BG ⌟ , f0 (inc p) ∙ f0 (inc q))
    trace p q =
      ⌞ πₙ₊₁ 1 (Σ¹ BG) ⌟ , inc (p ∙ q)         ≃∙⟨ n-Tr-set , refl ⟩≃∙
      ⌞ n-Tr∙ (Ωⁿ 2 (Σ¹ BG)) 2 ⌟ , inc (p ∙ q) ≃∙⟨ n-Tr-Ω¹ _ 1 .fst , n-Tr-Ω¹-∙ _ 1 p q ⟩≃∙
      ⌞ Ω¹ (n-Tr∙ (Ω¹ (Σ¹ BG)) 3) ⌟ , _        ≃∙⟨ Ω¹-ap ΩΣG≃∙G .fst , Ω¹-map-∙ (Equiv∙.to∙ ΩΣG≃∙G) (n-Tr-Ω¹ _ 1 · inc p) (n-Tr-Ω¹ _ 1 · inc q) ⟩≃∙
      ⌞ Ω¹ BG ⌟ , f0 (inc p) ∙ f0 (inc q)      ≃∙∎

  Ω²ΣG≃ΩG-inv l = trace .snd where
    trace : (⌞ Ω¹ BG ⌟ , l) ≃∙ (⌞ πₙ₊₁ 1 (Σ¹ BG) ⌟ , inc (Ω¹-map (suspend∙ BG) · l))
    trace =
      ⌞ Ω¹ BG ⌟                     , l                                ≃∙⟨ Ω¹-ap ΩΣG≃∙G .fst e⁻¹ , (Ω¹-ap-inv ΩΣG≃∙G ·ₚ l) ∙ ap (λ x → Ω¹-map x · l) ΩΣG≃∙G-inv ⟩≃∙
      ⌞ Ω¹ (n-Tr∙ (Ω¹ (Σ¹ BG)) 3) ⌟ , Ω¹-map (inc∙ ∘∙ suspend∙ BG) · l ≃∙˘⟨ n-Tr-Ω¹ _ 1 .fst , (n-Tr-Ω¹-inc _ 1 ·ₚ _) ∙ (Ω¹-map-∘ inc∙ (suspend∙ BG) ·ₚ l) ⟩≃∙˘
      ⌞ n-Tr∙ (Ωⁿ 2 (Σ¹ BG)) 2 ⌟    , inc (Ω¹-map (suspend∙ BG) · l)   ≃∙˘⟨ n-Tr-set , refl ⟩≃∙˘
      ⌞ πₙ₊₁ 1 (Σ¹ BG) ⌟            , inc (Ω¹-map (suspend∙ BG) · l)   ≃∙∎

π₂ΣG≅ΩG : πₙ₊₁ 1 (Σ¹ BG) Groups.≅ π₁Groupoid.π₁ BG (grp .has-is-groupoid)
π₂ΣG≅ΩG = total-iso Ω²ΣG≃ΩG Ω²ΣG≃ΩG-pres

The Hopf fibration🔗

We can now prove that the total space of the Hopf fibration defined above is the join of $G$ with itself.

join→hopf : (G ∗ G) → Σ _ Hopf
join→hopf (inl x) = north , x
join→hopf (inr x) = south , x
join→hopf (join a b i) = record
  { fst = merid (a \\ b) i
  ; snd = attach (∂ i) (λ { (i = i0) → a ; (i = i1) → b })
    (inS (μ-\\-l a b i))
  }

join→hopf-split : ∀ x p → fibre join→hopf (x , p)
join→hopf-split = Susp-elim _
  (λ p → inl p , refl)
  (λ p → inr p , refl)
  (λ x i → filler x i i1)

  module surj where module _ (x : fst BG) where
    coh : ∀ a → PathP
      (λ i → fibre join→hopf (merid x i , ua-inc (μr x) a i))
      (inl a , refl) (inr (μ a x) , refl)
    coh a i .fst = join a (μ a x) i
    coh a i .snd j .fst = merid (μ-\\-r a x j) i
    coh a i .snd j .snd = attach (∂ i)
      (λ { (i = i0) → a ; (i = i1) → μ a x })
      (inS (∨-square (μ-zig a x) j i))

    open ua→ {e = μr x} {B = λ i z → fibre join→hopf (merid x i , z)} {f₀ = λ p → inl p , refl} {f₁ = λ p → inr p , refl} coh public

hopf→join : (Σ _ Hopf) → G ∗ G
hopf→join a = uncurry join→hopf-split a .fst

hopf→join→hopf : is-right-inverse hopf→join join→hopf
hopf→join→hopf a = uncurry join→hopf-split a .snd

join→hopf→join : is-left-inverse hopf→join join→hopf
join→hopf→join (inl x) = refl
join→hopf→join (inr x) = refl
join→hopf→join (join a b i) =
  let it = attach (∂ i) (λ { (i = i0) → a ; (i = i1) → b }) (inS (μ-\\-l a b i)) in
  comp (λ l → surj.filler (a \\ b) i l it .fst ≡ join a b i) (∂ i) λ where
    j (j = i0) → λ k → join a (μ-\\-l a b (i ∨ k)) (~ k ∨ i)
    j (i = i0) → λ k → join a (μ-\\-l a b k) (~ j ∧ ~ k)
    j (i = i1) → λ k → inr b

∫Hopf≃join : Σ _ Hopf ≃ (⌞ G ⌟ ∗ ⌞ G ⌟)
∫Hopf≃join .fst = hopf→join
∫Hopf≃join .snd = is-iso→is-equiv (iso join→hopf join→hopf→join hopf→join→hopf)