Skip to content
Permalink
Branch: master
Find file Copy path
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
78 lines (63 sloc) 3.68 KB
{-# OPTIONS --type-in-type --without-K #-}
open import lib.Prelude
import homotopy.FreudenthalIteratedSuspension1
open import homotopy.Pi1S1
open import homotopy.HStructure
open import homotopy.Pi2HSusp
open import homotopy.PiLessOfConnected
open Int
open Truncation
open LoopSpace
open Suspension
module homotopy.PiSNSusp where
open NSphereSusp
H-S¹ : H-Structure S¹.S¹ S¹.base
H-S¹ = record { _⊙_ = mult;
unitl = λ _ id;
unitr = S¹.S¹-elim _ id ((((!-inv-r S¹.loop ∘ ap (λ x S¹.loop ∘ ! x) red) ∘
ap (λ x x ∘ ! (ap (λ z mult z S¹.base) S¹.loop))
(ap-id S¹.loop)) ∘
ap (λ x ap (λ x' x') S¹.loop ∘ x)
(∘-unit-l (! (ap (λ z mult z S¹.base) S¹.loop)))) ∘ transport-Path (λ z mult z S¹.base) (λ z z) S¹.loop id);
unitcoh = id;
isequivl = S¹.S¹-elim _ (snd id-equiv) (HProp-unique (IsEquiv-HProp (λ x x)) _ _) } where
mutual
mult : S¹.S¹ -> S¹.S¹ -> S¹.S¹
mult = S¹.S¹-rec (λ x x) mult-loop
mult-hom = (S¹.S¹-elim _ S¹.loop ((!-inv-r-back S¹.loop S¹.loop ∘ ap (λ x x ∘ S¹.loop ∘ ! x) (ap-id S¹.loop)) ∘ transport-Path (λ x x) (λ x x) S¹.loop S¹.loop))
mult-loop = (λ≃ mult-hom)
red : (ap (λ z mult z S¹.base) S¹.loop) ≃ S¹.loop
red = ((Π≃β mult-hom {S¹.base}) ∘ ap (λ x ap≃ x {S¹.base}) (S¹.βloop/rec (\ x -> x) (mult-loop))) ∘ ap-o (λ f f S¹.base) mult S¹.loop
module SS = homotopy.FreudenthalIteratedSuspension1 (S¹.S¹) S¹.base (S^-Connected 0)
π1[Sn+1] : (n : Positive) -> π One (S^ (n +1)) (base^ (n +1)) ≃ Unit
π1[Sn+1] n = π1Connected≃Unit (tlp n) (S^ (S n)) (base^ (S n)) (SS.Susp'^-Connected'' n) (1<=pos n)
πk[Sn]-less : (k n : Positive) tlp k <tl tlp n π k (S^ n) (base^ n) ≃ Unit
πk[Sn]-less One One (ltSR (ltSR (ltSR ())))
πk[Sn]-less One (S n) lt = π1[Sn+1] n
πk[Sn]-less (S k) One lt = Sums.abort (pos-not-<=0 k (Inl (lt-unS lt)))
πk[Sn]-less (S k) (S n) lt = π (k +1) (S^ (n +1)) (base^ (n +1)) ≃〈 ! (SS.Stable.stable k n (k<=n->k<=2n-2 k n (Inl (lt-unS lt)))) 〉
π k (S^ n) (base^ n) ≃〈 πk[Sn]-less k n (lt-unS lt) 〉
Unit ∎
πn[Sn]-is-Int : n π n (S^ n) (base^ n) ≃ Int
πn[Sn]-is-Int One = π₁[S¹]-is-Int
πn[Sn]-is-Int (S One) = π₁[S¹]-is-Int ∘ π2Susp S¹.S¹ S¹.base S¹-is-Gpd (S^-Connected 0) H-S¹
πn[Sn]-is-Int (S (S n)) = πn[Sn]-is-Int (S n) ∘ ! (SS.Stable.stable (S n) (S n) (k<=n->k<=2n-2 (S n) (S n) (Inr (id , >pos->1 n (S n) ltS))))
-- consequences of stablity for k <= 2n - 2
-- n = 1: k <= 0
-- n = 2: k <= 2 : pi_1(S^2) = pi_2(S^3) and pi_2(S^2) = pi_3(S^3)
-- n = 3: k <= 4 : pi_1(S^3) = pi_2(S^4) and pi_2(S^3) = pi_3(S^4) and pi_3(S^3) = pi_4(s^4) and pi_4(S^3) = pi_5(S^4)
-- n = 4: k <= 6 : pi_1(S^4) = pi_2(S^5) and pi_2(S^4) = pi_3(S^5) and pi_3(S^4) = pi_4(s^5) and pi_4(S^4) = pi_5(s^5) and pi_5(S^4) = pi_6(s^5) and pi_6(S^4) = pi_7(S^5)
-- so:
-- k<n : pi_1(S^2) = pi_2(S^3) = pi_3(S^4) = pi_4(s^5) = ...
-- pi_1(S^3) = pi_2(S^4) = pi_3(S^5) = ...
-- pi_1(S^4) = pi_2(S^5)
-- k=n : pi_2(S^2) = pi_3(S^3) = pi_4(s^4) = pi_5(s^5) = ...
-- k>n : pi_4(S^3) = pi_5(S^4) = pi_6(s^5) = ...
-- pi_6(S^4) = pi_7(S^5)
-- to start the diagonals, can prove:
-- pi_1(S^2)
-- pi_1(S^3)
-- pi_1(S^4)
-- pi_2(S^2)
-- pi_4(S^3)
-- pi_6(S^4)
You can’t perform that action at this time.