nLab
relative homology

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Homological algebra

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nonabelian homological algebra

Context

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Contents

Definition

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.

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)) \,.
Remark

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”.

Properties

Long exact sequences

Proposition

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).

Proof

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.

Proposition

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 \,.
Proof

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.

Excision

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.

Proposition

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.

Proposition

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).

Remark

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

Example

For XX a CW complex, the inclusion of any subcomplex XXX' \hookrightarrow X is a good pair.

This is discussed at CW complex – Subcomplexes.

Proposition

If AXA \hookrightarrow X is a topological subspace inclusion which is good in the sense of def. 3, 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).

Proof

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. 2.

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. 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 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. 5 for AA a point.

Examples

Basic examples

Example

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}) \,.
Proof

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. 3. Therefore the example follows with prop. 5.

Detecting homology isomorphisms

Example

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.

Proof

Under the given assumotion the long exact sequence in prop. 1 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.

Proposition

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).

Proof

The inclusion X k1X kX_{k-1} \hookrightarrow X_k is clearly a good pair in the sense of def. 3. 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. 5 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.

References

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

Revised on October 4, 2013 06:54:11 by Anonymous (186.105.173.118)