nLab relative homology



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


Algebraic topology



In singular homology

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


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)) \,.

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


Long exact sequences


Let AiXA \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 n1H n1(A)H n1(i)H n1(X)H n1(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 cC n+1(X)c \in C_{n+1}(X), to the class represented by the boundary XcC n(A)C n(X)\partial^X c \in C_n(A) \hookrightarrow C_n(X).


This is the homology long exact sequence induced by the given short exact sequence 0C (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.


Let BAXB \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 n1(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 \,.

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

0C (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.


Let ZAXZ \hookrightarrow A \hookrightarrow X be a sequence of topological subspace inclusions such that the closure Z¯\bar Z of ZZ is still contained in the interior A A^\circ of AA: Z¯A \bar Z \hookrightarrow A^\circ.


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

H n(XZ,AZ)H n(X,A) H_n(X-Z, A-Z) \stackrel{\simeq}{\to} H_n(X,A)

for all nn \in \mathbb{N}.

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


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

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

for all nn \in \mathbb{N}.

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


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

Homotopy invariance


Relative homology is homotopy invariant in both arguments.


Relation to reduced homology of quotient topological spaces


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

  1. AA is closed inside XX;

  2. AA has an neighbourhood in XX which is a deformation retract of AA.


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

This is discussed at CW complex – Subcomplexes.


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

H n(X,A)H˜ n(X/A). H_n(X , A) \simeq \tilde H_n(X/A) \,.

For instance (Hatcher, prop. 2.22).


By assumption we can find a neighbourhood AjUXA \stackrel{j}{\to} U \hookrightarrow X such that AUA \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 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) (XA,UA) (X/A,A/A) (id,j/A) (X/A,U/A) (X/AA/A,U/AA/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 nn \in \mathbb{N} the commuting diagram of abelian groups

H n(X,A) H n(id,j) H n(X,U) H n(XA,UA) H n(X/A,A/A) H n(id,j/A) H n(X/A,U/A) H n(X/AA/A,U/AA/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).

Relation to reduced homology


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) \,.

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


Basic examples


The reduced singular homology of the nn-sphere S nS^{n} equals the S n1S^{n-1}-relative homology of the nn-disk with respect to the canonical boundary inclusion S n1D nS^{n-1} \hookrightarrow D^n: for all nn \in \mathbb{N}

H˜ (S n)H (D n,S n1). \tilde H_\bullet(S^n) \simeq H_\bullet(D^n, S^{n-1}) \,.

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

S nD n/S n1. 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


If an inclusion AXA \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.


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

0H 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.

Relative homology of CW-complexes

Let XX be a CW-complex and write

X 0X 1X 2X X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X

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


The relative singular homology of the filtering degrees is

H n(X k,X k1){[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) nSetCells(X)_n \in Set denotes the set of nn-cells of XX and [Cells(X) n]\mathbb{Z}[Cells(X)_n] denotes the free abelian group on this set.

For instance (Hatcher, lemma 2.34).


The inclusion X k1X kX_{k-1} \hookrightarrow X_k is clearly a good pair in the sense of def. . The quotient X k/X k1X_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 k1)H˜ n(X k/X k1)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.


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

Last revised on June 8, 2022 at 17:43:18. See the history of this page for a list of all contributions to it.