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relative homology

Definition

In singular homology

Let $X$ be a topological space and $A\hookrightarrow X$ a subspace. Write $C_{•}(X)$ for the chain complex of singular homology on $X$ and $C_{•}(A)\hookrightarrow C_{•}(X)$ for the chain map induced by the subspace inclusion.

Definition

The cokernel of this inclusion, hence the quotient $C_{•}(X)/C_{•}(A)$ of $C_{•}(X)$ by the image of $C_{•}(A)$ under the inclusion, is the chain complex of $A$-relative singular chains.

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

• a cycle is called an $A$-relative singular cycle.

• The chain homology of the quotient is the $A$-relative singular homology of $X$

  $$H_n(X,A)≔H_n(C_{•}(X)/C_{•}(A)).$$H_n(X , A)\coloneqq H_n(C_\bullet(X)/C_\bullet(A)) \,.

Remark

This means that a singular $(n+1)$-chain $c\in C_{n+1}(X)$ is an $A$-relative cycle if its boundary $\partial c\in C_n(X)$ is, while not necessarily 0, contained in the $n$-chains of $A$: $\partial c\in C_n(A)\hookrightarrow C_n(X)$. So it vanishes only “up to contributions coming from $A$”.

Properties

Long exact sequences

Proposition

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

$$\cdots \to H_n(A)\stackrel{H_n(i)}{\to }{H}_n(X)\to H_n(X,A)\stackrel{{\delta }_{n-1}}{\to }{H}_{n-1}(A)\stackrel{H_{n-1}(i)}{\to }{H}_{n-1}(X)\to H_{n-1}(X,A)\to \cdots .$$\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 ${\delta }_n:H_{n+1}(X,A)\to H_n(A)$ sends an element $[c]\in H_{n+1}(X,A)$ represented by an $A$-relative cycle $c\in C_{n+1}(X)$, to the class represented by the boundary ${\partial }^{X}c\in C_n(A)\hookrightarrow C_n(X)$.

Proof

This is the homology long exact sequence induced by the given short exact sequence $0\to C_{•}(A)\stackrel{i}{\hookrightarrow }{C}_{•}(X)\to \mathrm{coker}(i)\simeq C_{•}(X)/C_{•}(A)\to 0$ of chain complexes.

Proposition

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

$$\cdots \to H_n(A,B)\to H_n(X,B)\to H_n(X,A)\to H_{n-1}(A,B)\to \cdots .$$\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

$$0\to C_{•}(A)/C_{•}(B)\to C_{•}(X)/C_{•}/B)\to C_{•}(X)/C_{•}(A)\to 0.$$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.

Excision

Let $Z\hookrightarrow A\hookrightarrow X$ be a sequence of topological subspace inclusions such that the closure $\overline{Z}$ of $Z$ is still contained in the interior $A^{\circ }$ of $A$: $\overline{Z}\hookrightarrow A^{\circ }$.

Proposition

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

$$H_n(X-Z,A-Z)\stackrel{\simeq }{\to }{H}_n(X,Z)$$H_n(X-Z, A-Z) \stackrel{\simeq}{\to} H_n(X,Z)

for all $n\in \mathbb{N}$.

Let $A,B\hookrightarrow X$ be two topological subspaces such that their interior is a cover $A^{\circ }\coprod B^{\circ }\to X$ of $X$.

Proposition

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

$$H_n(B,A\cap B)\stackrel{\simeq }{\to }{H}_n(X,A)$$H_n(B, A \cap B) \stackrel{\simeq}{\to} H_n(X,A)

for all $n\in \mathbb{N}$.

A proof is spelled out in (Hatcher, from p. 128 on).

Remark

These two propositions are equivalent to each other. To see this identify $B=X-Z$.

Homotopy invariance

Definition

Relative homology is homotopy invariant in both arguments.

(…)

Relation to reduced homology of quotient topological spaces

Definition

A topological subspace inclusion $A\hookrightarrow X$ is called a good pair if

1. $A$ is closed inside $X$;

2. $A$ has an neighbourhood in $X$ which is a deformation retract of $A$.

Example

For $X$ a CW complex, the inclusion of any subcomplex $X\prime \hookrightarrow X$ is a good pair.

This is discussed at CW complex – Subcomplexes.

Proposition

If $A\hookrightarrow X$ is a topological subspace inclusion which is good in the sense of def. 3, then the $A$-relative singular homology of $X$ coincides with the reduced singular homology of the quotient space $X/A$:

$$H_n(X/A)\simeq {\tilde{H}}_n(X,A).$$H_n(X/A) \simeq \tilde H_n(X , A) \,.

For instance (Hatcher, prop. 2.22).

Proof

By assumption we can find a neighbourhood $A\stackrel{j}{\to }U\hookrightarrow X$ such that $A\hookrightarrow U$ 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 $n\in \mathbb{N}$ the canonical morphism $H_n(X,A)\stackrel{H_n(\mathrm{id},j)}{\to }{H}_n(X,U)$ is an isomorphism, by prop. 2.

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

$$\begin{array}{ccccc}(X,A)& \stackrel{(\mathrm{id},j)}{\to }& (X,U)& \leftarrow & (X-A,U-A)\\ \downarrow & & \downarrow & & {\downarrow }^{\simeq }\\ (X/A,A/A)& \stackrel{(\mathrm{id},j/A)}{\to }& (X/A,U/A)& \leftarrow & (X/A-A/A,U/A-A/A)\end{array}.$$\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 $n\in \mathbb{N}$ the commuting diagram of abelian groups

$$\begin{array}{ccccc}{H}_n(X,A)& \underset{\simeq }{\overset{H_n(\mathrm{id},j)}{\to }}& H_n(X,U)& \stackrel{\simeq }{\leftarrow }& H_n(X-A,U-A)\\ \downarrow & & \downarrow & & {\downarrow }^{\simeq }\\ H_n(X/A,A/A)& \underset{\simeq }{\overset{H_n(\mathrm{id},j/A)}{\to }}& H_n(X/A,U/A)& \stackrel{\simeq }{\leftarrow }& H_n(X/A-A/A,U/A-A/A)\end{array}.$$\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. 3 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).

Relation to reduced homology

Proposition

Let $X$ be a inhabited topological space and let $x:*\hookrightarrow X$ any point. Then the relative singular homology $H_n(X,*)$ is isomorphic to the absolute reduced singular homology ${\tilde{H}}_n(X)$ of $X$

$$H_n(X,*)\simeq {\tilde{H}}_n(X).$$H_n(X , *) \simeq \tilde H_n(X) \,.

Proof

This is the special case of prop. 5 for $A$ a point.

Examples

Basic examples

Example

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

$${\tilde{H}}_{•}(S^n)\simeq H_{•}(D^n,S^{n-1}).$$\tilde H_\bullet(S^n) \simeq H_\bullet(D^n, S^{n-1}) \,.

Proof

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

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

Moreover the boundary inclusion is evidently a good pair in the sense of def. 3. Therefore the example follows with prop. 5.

Detecting homology isomorphisms

Example

If an inclusion $A\hookrightarrow X$ is such that all relative homology vanishes, $H_{•}(X,A)\simeq 0$, then the inclusion induces isomorphisms on all singular homology groups.

Proof

Under the given assumotion the long exact sequence in prop. 1 secomposes into short exact pieces of the form

$$0\to H_n(A)\to H_n(X)\to 0.$$0 \to H_n(A) \to H_n(X) \to 0 \,.

Exactness says that the middle morphism here is an isomorphism.

Relative homology of CW-complexes

Let $X$ be a CW-complex and write

$$X_0\hookrightarrow X_1\hookrightarrow X_2\hookrightarrow \cdots \hookrightarrow X$$X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X

for its filtered topological space-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells.

Proposition

The relative singular homology of the filtering degrees is

$$H_n(X_k,X_{k-1})\simeq \{\begin{array}{cc}\mathbb{Z}[\mathrm{Cells}(X{)}_n]& \mathrm{if}k=n\\ 0& \mathrm{otherwise}\end{array},$$H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[Cells(X)_n] & if\; k = n \\ 0 & otherwise } \right. \,,

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

For instance (Hatcher, lemma 2.34).

Proof

The inclusion $X_{k-1}\hookrightarrow X_k$ is clearly a good pair in the sense of def. 3. The quotient $X_k/X_{k-1}$ is by definition of CW-complexes a wedge sum of $k$-spheres, one for each element in $\mathrm{kCell}$. Therefore by prop. 5 we have an isomorphism $H_n(X_k,X_{k-1})\simeq {\tilde{H}}_n(X_k/X_{k-1})$ 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.

Related concepts

• relative cohomology

• cellular homology

References

A standard textbook account for relative singular homology is section 2.1 of

• Allen Hatcher, Algebraic Topology