This commit is contained in:
Michael Zhang 2025-01-24 05:46:08 -06:00
parent c7f6942d7c
commit 4e65fc2a32
2 changed files with 39 additions and 11 deletions
src/ThesisWork/HomotopyGroupLES2

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@ -13,7 +13,7 @@ open import Cubical.Structures.Successor
open import ThesisWork.Exactness
open import ThesisWork.HomotopyGroupLES2.Util
open import ThesisWork.HomotopyGroupLES2.Step1
open import ThesisWork.HomotopyGroupLES2.Step1 renaming (A to A' ; f^ to f^')
open import ThesisWork.HomotopyGroupLES2.Lemma415
private
@ -23,6 +23,8 @@ private
pattern 3+ n = suc (suc (suc n))
module _ {X @ (X , x) : Pointed } {Y∙ @ (Y , y) : Pointed } (f∙ @ (f , f₀) : X∙ Y∙) where
A = A' f∙
f^ = f^' f∙
F∙ = Fiber f∙
B : (n : ) Pointed
@ -37,28 +39,50 @@ module _ {X∙ @ (X , x) : Pointed } {Y∙ @ (Y , y) : Pointed } (f∙ @ (
g^ 2 = δ f∙
g^ (3+ n) = (g^ n)
_ : A f∙ 3 B 3
_ : A 3 B 3
_ = ua∙ (eqv∙ f∙ .fst) (eqv∙ f∙ .snd)
module Lemma416 where
η : (n : ) A f∙ n ≃∙ B n
η : (n : ) A n ≃∙ B n
η 0 = idEquiv∙ Y∙
η 1 = idEquiv∙ X∙
η 2 = idEquiv∙ F∙
η (3+ n) =
A f∙ (3+ n) ≃∙⟨ eqv∙ (f^ f∙ n)
Ω (A f∙ n) ≃∙⟨ cong-≃∙ Ω (η n)
A (3+ n) ≃∙⟨ eqv∙ (f^ n)
Ω (A n) ≃∙⟨ cong-≃∙ Ω (η n)
Ω (B n) ∎≃∙
eqvfg : (n : ) (≃∙map (η n) ∘∙ f^ f∙ n) ∙∼ (g^ n ∘∙ ≃∙map (η (suc n)))
eqvfg 0 = {! !}
eqvfg 1 = {! !}
eqvfg 2 = {! !}
eqvfg (3+ n) = {! !}
A≡B : A B
A≡B = funExt (λ n ua∙ (η n .fst) (η n .snd))
postulate
-- TODO: Finish
f∙g : (n : ) (≃∙map (η n) ∘∙ f^ n) P (g^ n ∘∙ ≃∙map (η (suc n)))
-- eqvfg 0 = {! !}
-- eqvfg 1 = {! !}
-- eqvfg 2 = {! !}
-- eqvfg (3+ n) = {! !}
-- TODO: Is this actually easier?
-- Potential point of deviation from FVD thesis
f≡g' : (n : ) PathP (λ i A≡B i (suc n) A≡B i n) (f^ n) (g^ n)
f≡g' 0 i = {! !}
-- at i = i0, A^ 1 →∙ A^ 0
-- at i = i1, B^ 1 →∙ B^ 0
f≡g' 1 i = {! !}
f≡g' 2 i = {! !}
f≡g' (3+ n) = {! !}
f≡g : PathP (λ i (n : ) A≡B i (suc n) A≡B i n) f^ g^
f≡g = funExt f≡g'
open Lemma416 using (f≡g)
BⁿisExact : (n : ) isExactAt∙ (g^ (suc n)) (g^ n)
BⁿisExact n = transp (λ i isExactAt∙ ((f≡g i) (suc n)) ((f≡g i) n)) i0 (AⁿisExact f∙ n)
open ExactSequence∙
BSeq : ExactSequence∙ +
BSeq .seq = B
BSeq .fun = g^
BSeq .exactness = {! !}
BSeq .exactness = BⁿisExact

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@ -1,3 +1,7 @@
{-# OPTIONS --cubical #-}
module ThesisWork.HomotopyGroupLES2.Step3 where
open import ThesisWork.HomotopyGroupLES2.Step2 renaming (B to B' ; g^ to g^')
module _ {X @ (X , x) : Pointed } {Y∙ @ (Y , y) : Pointed } (f∙ @ (f , f₀) : X∙ Y∙) where