relative homology in nLab

Context

Homological algebra

Algebraic topology

• Definition
• In singular homology
• Properties
• Long exact sequences
• Excision
• Homotopy invariance
• Relation to reduced homology of quotient topological spaces
• Relation to reduced homology
• Examples
• Basic examples
• Detecting homology isomorphisms
• Relative homology of CW-complexes
• Related concepts
• References

Definition

The cokernel of this inclusion, hence the quotient C •(X)/C •(A)C_\bullet(X)/C_\bullet(A) of C •(X)C_\bullet(X) by the image of C •(A)C_\bullet(A) under the inclusion, is the chain complex of AA-relative singular chains.

• A boundary in this quotient is called an AA-relative singular boundary,

• a cycle is called an AA-relative singular cycle.

• The chain homology of the quotient is the AA-relative singular homology of XX

  H n(X,A)≔H n(C •(X)/C •(A)). H_n(X , A)\coloneqq H_n(C_\bullet(X)/C_\bullet(A)) \,.

Proposition

Let A↪iXA \stackrel{i}{\hookrightarrow} X. The corresponding relative homology sits in a long exact sequence of the form

⋯→H n(A)→H n(i)H n(X)→H n(X,A)→δ n−1H n−1(A)→H n−1(i)H n−1(X)→H n−1(X,A)→⋯. \cdots \to H_n(A) \stackrel{H_n(i)}{\to} H_n(X) \to H_n(X, A) \stackrel{\delta_{n-1}}{\to} H_{n-1}(A) \stackrel{H_{n-1}(i)}{\to} H_{n-1}(X) \to H_{n-1}(X, A) \to \cdots \,.

The connecting homomorphism δ n:H n+1(X,A)→H n(A)\delta_{n} \colon H_{n+1}(X, A) \to H_n(A) sends an element [c]∈H n+1(X,A)[c] \in H_{n+1}(X, A) represented by an AA-relative cycle c∈C n+1(X)c \in C_{n+1}(X), to the class represented by the boundary ∂ Xc∈C n(A)↪C n(X)\partial^X c \in C_n(A) \hookrightarrow C_n(X).

Proof

This is the homology long exact sequence induced by the given short exact sequence 0→C •(A)↪iC •(X)→coker(i)≃C •(X)/C •(A)→00 \to C_\bullet(A) \stackrel{i}{\hookrightarrow} C_\bullet(X) \to coker(i) \simeq C_\bullet(X)/C_\bullet(A) \to 0 of chain complexes.

Proposition

Let B↪A↪XB \hookrightarrow A \hookrightarrow X be a sequence of two inclusions. Then there is a long exact sequence of relative homology groups of the form

⋯→H n(A,B)→H n(X,B)→H n(X,A)→H n−1(A,B)→⋯. \cdots \to H_n(A , B) \to H_n(X , B) \to H_n(X , A ) \to H_{n-1}(A , B) \to \cdots \,.

Proof

Observe that we have a (degreewise) short exact sequence of chain complexes

0→C •(A)/C •(B)→C •(X)/C •(B)→C •(X)/C •(A)→0. 0 \to C_\bullet(A)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(A) \to 0 \,.

The corresponding homology long exact sequence is the long exact sequence in question.

Proposition

In the above situation, the inclusion (X−Z,A−Z)↪(X,A)(X-Z, A-Z) \hookrightarrow (X,A) induces isomorphism in relative singular homology groups

H n(X−Z,A−Z)→≃H n(X,A) H_n(X-Z, A-Z) \stackrel{\simeq}{\to} H_n(X,A)

for all n∈ℕn \in \mathbb{N}.

Proposition

In the above situation, the inclusion (B,A∩B)↪(X,A)(B, A \cap B) \hookrightarrow (X,A) induces isomorphisms in relative singular homology groups

H n(B,A∩B)→≃H n(X,A) H_n(B, A \cap B) \stackrel{\simeq}{\to} H_n(X,A)

for all n∈ℕn \in \mathbb{N}.

Definition

Relative homology is homotopy invariant in both arguments.

Example

For XX a CW complex, the inclusion of any subcomplex A↪XA \hookrightarrow X is a good pair (called a CW-pair (X,A)(X,A)).

Proof

By assumption we can find a neighbourhood A→jU↪XA \stackrel{j}{\to} U \hookrightarrow X such that A↪UA \hookrightarrow U has a deformation retract and hence in particular is a homotopy equivalence and so induces also isomorphisms on all singular homology groups.

It follows in particular that for all n∈ℕn \in \mathbb{N} the canonical morphism H n(X,A)→H n(id,j)H n(X,U)H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U) is an isomorphism, by prop. .

Given such UU we have an evident commuting diagram of pairs of topological spaces

(X,A) →(id,j) (X,U) ← (X−A,U−A) ↓ ↓ ↓ ≃ (X/A,A/A) →(id,j/A) (X/A,U/A) ← (X/A−A/A,U/A−A/A). \array{ (X,A) &\stackrel{(id,j)}{\to}& (X,U) &\leftarrow& (X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ (X/A, A/A) &\stackrel{(id,j/A)}{\to}& (X/A, U/A) &\leftarrow& (X/A - A/A, U/A - A/A) } \,.

Here the right vertical morphism is in fact a homeomorphism.

Applying relative singular homology to this diagram yields for each n∈ℕn \in \mathbb{N} the commuting diagram of abelian groups

H n(X,A) →≃H n(id,j) H n(X,U) ←≃ H n(X−A,U−A) ↓ ↓ ↓ ≃ H n(X/A,A/A) →≃H n(id,j/A) H n(X/A,U/A) ←≃ H n(X/A−A/A,U/A−A/A). \array{ H_n(X,A) &\underoverset{\simeq}{H_n(id,j)}{\to}& H_n(X,U) &\stackrel{\simeq}{\leftarrow}& H_n(X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H_n(X/A, A/A) &\underoverset{\simeq}{H_n(id,j/A)}{\to}& H_n(X/A, U/A) &\stackrel{\simeq}{\leftarrow}& H_n(X/A - A/A, U/A - A/A) } \,.

Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).

Proposition

Let XX be a inhabited topological space and let x:*↪Xx \colon * \hookrightarrow X any point. Then the relative singular homology H n(X,*)H_n(X , *) is isomorphic to the absolute reduced singular homology H˜ n(X)\tilde H_n(X) of XX

H n(X,*)≃H˜ n(X). H_n(X , *) \simeq \tilde H_n(X) \,.

Proof

This is the special case of prop. for AA a point.

Example

The reduced singular homology of the nn-sphere S nS^{n} equals the S n−1S^{n-1}-relative homology of the nn-disk with respect to the canonical boundary inclusion S n−1↪D nS^{n-1} \hookrightarrow D^n: for all n∈ℕn \in \mathbb{N}

H˜ •(S n)≃H •(D n,S n−1). \tilde H_\bullet(S^n) \simeq H_\bullet(D^n, S^{n-1}) \,.

Proof

The nn-sphere is homeomorphic to the nn-disk with its entire boundary identified with a point:

S n≃D n/S n−1. S^n \simeq D^n/S^{n-1} \,.

Moreover the boundary inclusion is evidently a good pair in the sense of def. . Therefore the example follows with prop. .

Example

If an inclusion A↪XA \hookrightarrow X is such that all relative homology vanishes, H •(X,A)≃0H_\bullet(X , A) \simeq 0, then the inclusion induces isomorphisms on all singular homology groups.

Proof

Under the given assumotion the long exact sequence in prop. secomposes into short exact pieces of the form

0→H n(A)→H n(X)→0. 0 \to H_n(A) \to H_n(X) \to 0 \,.

Exactness says that the middle morphism here is an isomorphism.

Proposition

The relative singular homology of the filtering degrees is

H n(X k,X k−1)≃{ℤ[Cells(X) n] ifk=n 0 otherwise, H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[Cells(X)_n] & if\; k = n \\ 0 & otherwise } \right. \,,

where Cells(X) n∈SetCells(X)_n \in Set denotes the set of nn-cells of XX and ℤ[Cells(X) n]\mathbb{Z}[Cells(X)_n] denotes the free abelian group on this set.

Proof

The inclusion X k−1↪X kX_{k-1} \hookrightarrow X_k is clearly a good pair in the sense of def. . The quotient X k/X k−1X_k/X_{k-1} is by definition of CW-complexes a wedge sum of kk-spheres, one for each element in kCellkCell. Therefore by prop. we have an isomorphism H n(X k,X k−1)≃H˜ n(X k/X k−1)H_n(X_k , X_{k-1}) \simeq \tilde H_n( X_k / X_{k-1}) with the reduced homology of this wedge sum. The statement then follows by the respect of reduced homology for wedge sums as discussed at Reduced homology - Respect for wedge sums.