nLab relative homology

Definition

In singular homology

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

Definition

The cokernel of this inclusion, hence the quotient $C_\bullet(X)/C_\bullet(A)$ of $C_\bullet(X)$ by the image of $C_\bullet(A)$ 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 $X$

  $$H_n(X , A)\coloneqq H_n(C_\bullet(X)/C_\bullet(A)) \,.$$

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

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

The connecting homomorphism $\delta_{n} \colon H_{n+1}(X, A) \to H_n(A)$ 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 $0 \to C_\bullet(A) \stackrel{i}{\hookrightarrow} C_\bullet(X) \to coker(i) \simeq C_\bullet(X)/C_\bullet(A) \to 0$ of chain complexes.

Proposition

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

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

Proof

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

$$0 \to C_\bullet(A)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(A) \to 0 \,.$$

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 $\bar Z$ of $Z$ is still contained in the interior $A^\circ$ of $A$: $\bar Z \hookrightarrow A^\circ$.

Proposition

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

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

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

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

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 $A \hookrightarrow X$ is a good pair (called a CW-pair $(X,A)$).

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. , then the $A$-relative singular homology of $X$ coincides with the reduced singular homology of the quotient space $X/A$:

$$H_n(X , A) \simeq \tilde H_n(X/A) \,.$$

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 $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an isomorphism, by prop. .

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

$$\array{ (X,A) &\stackrel{(id,j)}{\to}& (X,U) &\leftarrow& (X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ (X/A, A/A) &\stackrel{(id,j/A)}{\to}& (X/A, U/A) &\leftarrow& (X/A - A/A, U/A - A/A) } \,.$$

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

$$\array{ H_n(X,A) &\underoverset{\simeq}{H_n(id,j)}{\to}& H_n(X,U) &\stackrel{\simeq}{\leftarrow}& H_n(X-A, U - A) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H_n(X/A, A/A) &\underoverset{\simeq}{H_n(id,j/A)}{\to}& H_n(X/A, U/A) &\stackrel{\simeq}{\leftarrow}& H_n(X/A - A/A, U/A - A/A) } \,.$$

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).

Relation to reduced homology

Proposition

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

$$H_n(X , *) \simeq \tilde H_n(X) \,.$$

Proof

This is the special case of prop. 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 $n \in \mathbb{N}$

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

Proof

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

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

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

Detecting homology isomorphisms

Example

If an inclusion $A \hookrightarrow X$ is such that all relative homology vanishes, $H_\bullet(X , A) \simeq 0$, 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

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

Exactness says that the middle morphism here is an isomorphism.

Relative homology of CW-complexes

Let $X$ be a CW-complex and write

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

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

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

where $Cells(X)_n \in Set$ denotes the set of $n$-cells of $X$ and $\mathbb{Z}[Cells(X)_n]$ 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. . The quotient $X_k/X_{k-1}$ is by definition of CW-complexes a wedge sum of $k$-spheres, one for each element in $kCell$. Therefore by prop. 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

• Gauss-Manin connection

References

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

• Allen Hatcher, Algebraic Topology