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Long exact sequence of a triple in singular homology

Statement

For spaces $B \subseteq A \subseteq X$ and every abelian group $G$ there is a long exact sequence $\dots \to H_{n} (A, B; G) \to H_{n} (X, B; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A, B; G) \to \dots,$ where the first two maps are induced by inclusions and $δ$ is the connecting homomorphism of the degreewise short exact sequence $0 \to C_{∙} (A, B; G) \to C_{∙} (X, B; G) \to C_{∙} (X, A; G) \to 0$ of relative singular chain complexes. Moreover $δ$ factors as the connecting map $H_{n} (X, A; G) \to H_{n - 1} (A; G)$ of the pair $(X, A)$ followed by the quotient map $H_{n - 1} (A; G) \to H_{n - 1} (A, B; G)$ .

Facts & Assumptions

Given: Spaces $B \subseteq A \subseteq X$ and an abelian group $G$ .

[F1]

The relative singular chain group is $C_{n} (X, A; G) = C_{n} (X; G) / C_{n} (A; G)$ , with $C_{n} (A; G) \subseteq C_{n} (X; G)$ induced by inclusion, and both $A = \emptyset$ and $A = X$ are admitted (Relative singular chain complex).

[F2]

The singular boundary descends to homomorphisms ${\overset{ˉ}{\partial}}_{n} : C_{n} (X, A; G) \to C_{n - 1} (X, A; G)$ with ${\overset{ˉ}{\partial}}_{n - 1} {\overset{ˉ}{\partial}}_{n} = 0$ (Boundary on relative chains).

[F3]

A short exact sequence of chain complexes is a sequence of chain maps $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ that is exact in each degree (Short exact sequence of complexes), a chain complex being a graded family with $d_{n - 1} d_{n} = 0$ (Chain complex in an abelian category).

[F4]

A short exact sequence of complexes in an abelian category induces a long exact sequence in homology with connecting maps $\partial_{n}$ (The long exact sequence in homology).

[L1]

$H_{n} (X, A; G)$ is by definition the homology of the complex $C_{∙} (X, A; G)$ (Relative singular homology).

[L2]

The connector of the pair sequence is fixed on cycles by $δ [c] : = [\partial c]$ for a relative cycle $c$ with $\partial c \in C_{n - 1} (A; G)$ ; the quotient map is the third arrow of the chain sequence (Relative connecting homomorphism on cycles).

Proof

technique · degreewise-quotient

1.1F1givenalgebra

In each degree the sequence $0 \to C_{n} (A, B; G) \to C_{n} (X, B; G) \to C_{n} (X, A; G) \to 0$ is the sequence of quotients $C_{n} (A) / C_{n} (B) \to C_{n} (X) / C_{n} (B) \to C_{n} (X) / C_{n} (A)$ induced by $C_{n} (B) \subseteq C_{n} (A) \subseteq C_{n} (X)$ by [F1]; it is exact because the first map is injective (the inclusion $C_{n} (A) \to C_{n} (X)$ descends injectively after dividing by the common subgroup $C_{n} (B)$ ), the second is surjective, and its kernel is exactly $C_{n} (A) / C_{n} (B)$ .

2.1F2F3step 1.1

By [F2] all three boundary maps descend to the quotients, so the degreewise maps of step 1.1 commute with the boundaries and form chain maps; hence they constitute a short exact sequence of complexes in the sense of [F3].

3.1F4L1step 2.1

Applying [F4] to the short exact sequence of step 2.1 gives the long exact sequence of the statement; the homology groups are $H_{n} (X, A; G)$ , $H_{n} (X, B; G)$ , $H_{n} (A, B; G)$ by [L1], and the first two maps are induced by inclusions because the chain maps of step 1.1 are.

4.1L2step 3.1algebra∎

For the factorization, let $c$ be a relative cycle for $(X, A)$ with $\partial c \in C_{n - 1} (A; G)$ , representing a class in $H_{n} (X, A; G)$ . The connecting map of the triple sequence sends its class to the class of $\partial c$ in $H_{n - 1} (A, B; G)$ : this is the same cycle formula as in [L2] read in the middle complex $C_{∙} (X, B; G)$ , where the role of the subspace is played by $A$ modulo $B$ . The pair connector of $(X, A)$ sends the same class to $[\partial c] \in H_{n - 1} (A; G)$ by [L2], and the quotient map $H_{n - 1} (A; G) \to H_{n - 1} (A, B; G)$ is induced by the quotient chain map; composing gives the triple connector. Hence $δ$ factors as the pair connector followed by the quotient map.

Remarks

The case $B = \emptyset$ . Then $H_{n} (X, B; G) = H_{n} (X; G)$ and $H_{n} (A, B; G) = H_{n} (A; G)$ are the absolute groups and the sequence is the ordinary long exact sequence of the pair; the factorization statement is vacuous, the quotient map being an isomorphism.
The case $A = B$ . Then the middle complex is $C_{∙} (X, B; G)$ and the sequence reads $\dots \to H_{n} (B, B; G) \to H_{n} (X, B; G) \overset{≅}{\to} H_{n} (X, B; G) \to H_{n - 1} (B, B; G) \to \dots$ , consistent with the vanishing of the two outer groups.
This is the exact sequence used to compare successive sublevel manifolds and to compute the connecting map of a handle stage.

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Sources

Allen Hatcher, Algebraic Topology, Sections 2.1-2.2 (relative homology and long exact sequences) (standard reference, not scraped)