{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.Freudenthal
module homotopy.IterSuspensionStable where
module Susp^StableSucc {i} (X : Ptd i) (cX : is-connected 0 (de⊙ X))
(k n : ℕ) (Skle : S k ≤ n *2) where
private
Skle' : ⟨ S k ⟩ ≤T ⟨ n ⟩₋₁ +2+ ⟨ n ⟩₋₁
Skle' = ≤T-trans (⟨⟩-monotone-≤ Skle) (inl (lemma n))
where lemma : (n : ℕ) → ⟨ n *2 ⟩ == ⟨ n ⟩₋₁ +2+ ⟨ n ⟩₋₁
lemma O = idp
lemma (S n') = ap S (ap S (lemma n')
∙ ! (+2+-βr ⟨ S n' ⟩₋₂ ⟨ S n' ⟩₋₂))
private
module F = FreudenthalIso
⟨ n ⟩₋₂ k Skle' (⊙Susp^ n X)
(transport (λ t → is-connected t (de⊙ (⊙Susp^ n X)))
(+2+0 ⟨ n ⟩₋₂) (⊙Susp^-conn n cX))
stable : πS (S k) (⊙Susp^ (S n) X) ≃ᴳ πS k (⊙Susp^ n X)
stable =
πS (S k) (⊙Susp^ (S n) X)
≃ᴳ⟨ πS-Ω-split-iso k (⊙Susp^ (S n) X) ⟩
πS k (⊙Ω (⊙Susp^ (S n) X))
≃ᴳ⟨ Ω^S-group-Trunc-fuse-diag-iso k (⊙Ω (⊙Susp^ (S n) X)) ⁻¹ᴳ ⟩
Ω^S-group k (⊙Trunc ⟨ S k ⟩ (⊙Ω (⊙Susp^ (S n) X))) Trunc-level
≃ᴳ⟨ F.iso ⁻¹ᴳ ⟩
Ω^S-group k (⊙Trunc ⟨ S k ⟩ (⊙Susp^ n X)) Trunc-level
≃ᴳ⟨ Ω^S-group-Trunc-fuse-diag-iso k (⊙Susp^ n X) ⟩
πS k (⊙Susp^ n X) ≃ᴳ∎