Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Reduced homology of CW complexes #1175

Open
wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Reduced homology of CW complexes #1175

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
68 changes: 60 additions & 8 deletions Cubical/CW/ChainComplex.agda
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -10,10 +10,13 @@ open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Function

open import Cubical.Data.Nat
open import Cubical.Data.Int
open import Cubical.Data.Bool
open import Cubical.Data.Fin.Inductive.Base
open import Cubical.Data.Fin.Inductive.Properties
open import Cubical.Data.Sigma

open import Cubical.HITs.S1
open import Cubical.HITs.Sn
open import Cubical.HITs.Pushout
open import Cubical.HITs.Susp
Expand All @@ -23,6 +26,7 @@ open import Cubical.HITs.SphereBouquet.Degree
open import Cubical.Algebra.Group.Base
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Algebra.AbGroup
open import Cubical.Algebra.AbGroup.Instances.FreeAbGroup
open import Cubical.Algebra.ChainComplex


Expand Down Expand Up @@ -63,6 +67,9 @@ module _ {ℓ} (C : CWskel ℓ) where
isoCofBouquet : cofibCW n C → SphereBouquet n (Fin An)
isoCofBouquet = Iso.fun (BouquetIso-gen n An αn (snd C .snd .snd .snd n))

isoCofBouquetInv : SphereBouquet n (Fin An) → cofibCW n C
isoCofBouquetInv = Iso.inv (BouquetIso-gen n An αn (snd C .snd .snd .snd n))

isoCofBouquetInv↑ : SphereBouquet (suc n) (Fin An+1) → cofibCW (suc n) C
isoCofBouquetInv↑ = Iso.inv (BouquetIso-gen (suc n) An+1 αn+1 (snd C .snd .snd .snd (suc n)))

Expand Down Expand Up @@ -179,14 +186,59 @@ module _ {ℓ} (C : CWskel ℓ) where
∂≡∂↑ : ∂ n ≡ ∂↑
∂≡∂↑ = bouquetDegreeSusp (pre∂ n)

-- augmentation map, in order to define reduced homology
module augmentation where
ε : Susp (cofibCW 0 C) → SphereBouquet 1 (Fin 1)
ε north = inl tt
ε south = inl tt
ε (merid (inl tt) i) = inl tt
ε (merid (inr x) i) = (push fzero ∙∙ (λ i → inr (fzero , loop i)) ∙∙ (λ i → push fzero (~ i))) i
ε (merid (push x i₁) i) with (C .snd .snd .snd .fst x)
ε (merid (push x i₁) i) | ()

εδ : ∀ (x : cofibCW 1 C) → (ε ∘ (suspFun (to_cofibCW 0 C)) ∘ (δ 1 C)) x ≡ inl tt
εδ (inl tt) = refl
εδ (inr x) i = (push fzero ∙∙ (λ i → inr (fzero , loop i)) ∙∙ (λ i → push fzero (~ i))) (~ i)
εδ (push a i) j = (push fzero ∙∙ (λ i → inr (fzero , loop i)) ∙∙ (λ i → push fzero (~ i))) (i ∧ (~ j))

preϵ : SphereBouquet 1 (Fin (preboundary.An 0)) → SphereBouquet 1 (Fin 1)
preϵ = ε ∘ (suspFun isoCofBouquetInv) ∘ isoSuspBouquetInv
where
open preboundary 0

opaque
preϵpre∂≡0 : ∀ (x : SphereBouquet 1 (Fin (preboundary.An+1 0))) → (preϵ ∘ preboundary.pre∂ 0) x ≡ inl tt
preϵpre∂≡0 x = cong (ε ∘ (suspFun isoCofBouquetInv))
(Iso.leftInv sphereBouquetSuspIso
(((suspFun isoCofBouquet) ∘ (suspFun (to_cofibCW 0 C)) ∘ (δ 1 C) ∘ isoCofBouquetInv↑) x))
∙ cong ε (aux (((suspFun (to_cofibCW 0 C)) ∘ (δ 1 C) ∘ isoCofBouquetInv↑) x))
∙ εδ (isoCofBouquetInv↑ x)
where
open preboundary 0
aux : ∀ (x : Susp (cofibCW 0 C)) → (suspFun (isoCofBouquetInv) ∘ (suspFun isoCofBouquet)) x ≡ x
aux north = refl
aux south = refl
aux (merid a i) j = merid (Iso.leftInv (BouquetIso-gen 0 An αn (snd C .snd .snd .snd 0)) a j) i

ϵ : AbGroupHom (ℤ[A 0 ]) (ℤ[Fin 1 ])
ϵ = bouquetDegree preϵ

opaque
ϵ∂≡0 : compGroupHom (∂ 0) ϵ ≡ trivGroupHom
ϵ∂≡0 = sym (bouquetDegreeComp (preϵ) (preboundary.pre∂ 0))
∙ cong bouquetDegree (funExt preϵpre∂≡0)
∙ bouquetDegreeConst _ _ _

open ChainComplex

CW-ChainComplex : ChainComplex ℓ-zero
chain CW-ChainComplex n = ℤ[A n ]
bdry CW-ChainComplex n = ∂ n
bdry²=0 CW-ChainComplex n = ∂∂≡0 n

-- Cellular homology
Hˢᵏᵉˡ : (n : ℕ) → Group₀
Hˢᵏᵉˡ n = homology n CW-ChainComplex
CW-AugChainComplex : ChainComplex ℓ-zero
chain CW-AugChainComplex (zero) = ℤ[Fin 1 ]
chain CW-AugChainComplex (suc n) = ℤ[A n ]
bdry CW-AugChainComplex (zero) = augmentation.ϵ
bdry CW-AugChainComplex (suc n) = ∂ n
bdry²=0 CW-AugChainComplex (zero) = augmentation.ϵ∂≡0
bdry²=0 CW-AugChainComplex (suc n) = ∂∂≡0 n

-- Reduced cellular homology
H̃ˢᵏᵉˡ : (n : ℕ) → Group₀
H̃ˢᵏᵉˡ n = homology n CW-AugChainComplex
Loading
Loading