reduced homology in nLab

Context

Homological algebra

• Idea
• Definition
• Reduced singular homology
• Properties
• Relation to ordinary homology
• Relation to relative homology
• Relation to wedge sums
• Examples
• For singular homology
• Related concepts
• References

Definition

The augmentation map is the homomorphism of abelian groups

ϵ:C 0(X)→ℤ \epsilon \colon C_0(X) \to \mathbb{Z}

which adds up all the coefficients of all 0-chains:

ϵ::∑ in iσ i↦∑ in i. \epsilon \colon \colon \sum_{i} n_i \sigma_i \mapsto \sum_i n_i \,.

Since the boundary of a 1-chain is in the kernel of this map, it constitutes a chain map

ϵ:C •(X)→ℤ, \epsilon \colon C_\bullet(X) \to \mathbb{Z} \,,

where now ℤ\mathbb{Z} is regarded as a chain complex concentrated in degree 0.

Definition

The reduced singular chain complex C˜ •(X)\tilde C_\bullet(X) of XX is the kernel of the augmentation map, the chain complex sitting in the short exact sequence

0→C˜ •(X)→C •(X)→ϵℤ→0. 0 \to \tilde C_\bullet(X) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0 \,.

The reduced singular homology H˜ •(X)\tilde H_\bullet(X) of XX is the chain homology of the reduced singular chain complex

H˜ •(X)≔H •(C˜ •(X)). \tilde H_\bullet(X) \coloneqq H_\bullet(\tilde C_\bullet(X)) \,.

Definition

The reduced singular homology of XX, denoted H˜ •(X)\tilde H_\bullet(X), is the chain homology of the augmented chain complex

⋯→C 2(X)→∂ 2C 1(X)→∂ 1C 0(X)→ϵℤ→0. \cdots \to C_2(X) \stackrel{\partial_2}{\to} C_1(X) \stackrel{\partial_1}{\to} C_0(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0 \,.

Proposition

For n∈ℕn \in \mathbb{N} there is an isomorphism

H n(X)≃{H˜ n(X) forn≥1 H˜ 0(X)⊕ℤ forn=0 H_n(X) \simeq \left\{ \array{ \tilde H_n(X) & for \; n \geq 1 \\ \tilde H_0(X) \oplus \mathbb{Z} & for\; n = 0 } \right.

Proof

The homology long exact sequence of the defining short exact sequence C˜ •(C)→C •(X)→ϵℤ\tilde C_\bullet(C) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z} is, since ℤ\mathbb{Z} here is concentrated in degree 0, of the form

⋯→H˜ n(X)→H n(X)→0→⋯→0→⋯→H˜ 1(X)→H 1(X)→0→H˜ 0(X)→H 0(X)→ϵℤ→0. \cdots \to \tilde H_n(X) \to H_n(X) \to 0 \to \cdots \to 0 \to \cdots \to \tilde H_1(X) \to H_1(X) \to 0 \to \tilde H_0(X) \to H_0(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0 \,.

Here exactness says that all the morphisms H˜ n(X)→H n(X)\tilde H_n(X) \to H_n(X) for positive nn are isomorphisms. Moreover, since ℤ\mathbb{Z} is a free abelian group, hence a projective object, the remaining short exact sequence

0→H˜ 0(X)→H 0(X)→ℤ→0 0 \to \tilde H_0(X) \to H_0(X) \to \mathbb{Z} \to 0

is split (as discussed there) and hence H 0(X)≃H˜ 0(X)⊕ℤH_0(X) \simeq \tilde H_0(X) \oplus \mathbb{Z}.

Proposition

For X=*X = * the point, the morphism

H 0(ϵ):H 0(X)→ℤ H_0(\epsilon) \colon H_0(X) \to \mathbb{Z}

is an isomorphism. Accordingly the reduced homology of the point vanishes in every degree:

H˜ •(*)≃0. \tilde H_\bullet(*) \simeq 0 \,.

Proof

By the discussion at Singular homology – Relation to homotopy groups we have that

H n(*)≃{ℤ forn=0 0 otherwise. H_n(*) \simeq \left\{ \array{ \mathbb{Z} & for \; n = 0 \\ 0 & otherwise } \right. \,.

Moreover, it is clear that ϵ:C 0(*)→ℤ\epsilon \colon C_0(*) \to \mathbb{Z} is the identity map.

Proof

Consider the sequence of topological subspace inclusions

∅↪*↪xX. \emptyset \hookrightarrow * \stackrel{x}{\hookrightarrow} X \,.

By the discussion at Relative homology - long exact sequences this induces a long exact sequence of the form

⋯→H n+1(*)→H n+1(X)→H n+1(X,*)→H n(*)→H n(X)→H n(X,*)→⋯→H 1(X)→H 1(X,*)→H 0(*)→H 0(x)H 0(X)→H 0(X,*)→0. \cdots \to H_{n+1}(*) \to H_{n+1}(X) \to H_{n+1}(X,*) \to H_n(*) \to H_n(X) \to H_n(X,*) \to \cdots \to H_1(X) \to H_1(X,*) \to H_0(*) \stackrel{H_0(x)}{\to} H_0(X) \to H_0(X,*) \to 0 \,.

Here in positive degrees we have H n(*)≃0H_n(*) \simeq 0 and therefore exactness gives isomorphisms

H n(X)→≃H n(X,*)∀ n≥1 H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}

and hence with prop. isomorphisms

H˜ n(X)→≃H n(X,*)∀ n≥1. \tilde H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1} \,.

It remains to deal with the case in degree 0. To that end, observe that H 0(x):H 0(*)→H 0(X)H_0(x) \colon H_0(*) \to H_0(X) is a monomorphism: for this notice that we have a commuting diagram

H 0(*) →id H 0(*) H 0(x)↓ H 0(f)↗ ↓ ≃ H 0(ϵ) H 0(X) →H 0(ϵ) ℤ, \array{ H_0(*) &\stackrel{id}{\to}& H_0(*) \\ {}^{\mathllap{H_0(x)}}\downarrow &{}^{\mathllap{H_0(f)}}\nearrow& \downarrow^{\mathrlap{H_0(\epsilon)}}_\simeq \\ H_0(X) &\stackrel{H_0(\epsilon)}{\to}& \mathbb{Z} } \,,

where f:X→*f \colon X \to * is the terminal map. That the outer square commutes means that H 0(ϵ)∘H 0(x)=H 0(ϵ)H_0(\epsilon) \circ H_0(x) = H_0(\epsilon) and hence the composite on the left is an isomorphism. This implies that H 0(x)H_0(x) is an injection.

Therefore we have a short exact sequence as shown in the top of this diagram

0 → H 0(*) ↪H 0(x) H 0(X) → H 0(X,*) → 0 ≃↘ ↓ H 0(ϵ) ℤ. \array{ 0 &\to& H_0(*) &\stackrel{H_0(x)}{\hookrightarrow}& H_0(X) &\stackrel{}{\to}& H_0(X,*) &\to& 0 \\ && & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{H_0(\epsilon)}} & \\ && && \mathbb{Z} } \,.

Using this we finally compute

H˜ 0(X) ≔kerH 0(ϵ) ≃coker(H 0(x)) ≃H 0(X,*). \begin{aligned} \tilde H_0(X) & \coloneqq ker H_0(\epsilon) \\ & \simeq coker( H_0(x) ) \\ & \simeq H_0(X,*) \end{aligned} \,.

Proposition

For each n∈ℕn \in \mathbb{N} the homomorphism

(H˜ n(ι i)) i:⊕ iH˜ n(X i)→H˜ n(∨ iX i) (\tilde H_n(\iota_i))_i \colon \oplus_i \tilde H_n(X_i) \to \tilde H_n(\vee_i X_i)

is an isomorphism.

Example

The reduced singular homology of the 0-sphere S 0≃*∐*S^0 \simeq {*} \coprod {*} is

H˜ n(S 0)≃{ℤ ifn=0 0 otherwise. \tilde H_n(S^0) \simeq \left\{ \array{ \mathbb{Z} & if \; n = 0 \\ 0 & otherwise } \right. \,.