The Long Exact Sequence of a Pair

The long exact sequence of a pair is the fundamental computational tool in homology theory, relating the homology of a space, a subspace, and the relative homology groups.

Relative Homology

Definition

Let $(X, A)$ be a pair of spaces with $A \subseteq X$ . The relative chain group $C_n(X, A)$ is the quotient $C_n(X) / C_n(A)$ . The boundary map $\partial_n : C_n(X) \to C_{n-1}(X)$ descends to a map $\bar{\partial}_n : C_n(X, A) \to C_{n-1}(X, A)$ , yielding the relative homology groups $H_n(X, A) = \ker \bar{\partial}_n / \operatorname{im} \bar{\partial}_{n+1}.$

Relative homology captures the homological features of $X$ that are not present in $A$ . One should think of $H_n(X, A)$ as measuring $n$ -dimensional holes in $X$ modulo those already in $A$ .

Definition

The connecting homomorphism $\partial_* : H_n(X, A) \to H_{n-1}(A)$ is defined as follows. Given $[\alpha] \in H_n(X, A)$ represented by a relative cycle $\alpha \in C_n(X)$ with $\partial_n \alpha \in C_{n-1}(A)$ , set $\partial_*([\alpha]) = [\partial_n \alpha] \in H_{n-1}(A)$ .

The Long Exact Sequence

Theorem6.5Long Exact Sequence of a Pair

For any pair $(X, A)$ of topological spaces, there is a long exact sequence $\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial_*} H_{n-1}(A) \to \cdots \to H_0(X, A) \to 0$ where $i_*$ is induced by inclusion $i : A \hookrightarrow X$ and $j_*$ is induced by the quotient map $j : C_n(X) \to C_n(X,A)$ .

ExampleHomology of spheres

Using the pair $(D^n, S^{n-1})$ and the fact that $D^n$ is contractible, the long exact sequence gives $0 = H_k(D^n) \to H_k(D^n, S^{n-1}) \xrightarrow{\partial_*} H_{k-1}(S^{n-1}) \xrightarrow{i_*} H_{k-1}(D^n) = 0$ for $k \geq 2$ , yielding $H_k(D^n, S^{n-1}) \cong H_{k-1}(S^{n-1})$ . Since $(D^n, S^{n-1})$ is a good pair with $D^n/S^{n-1} \cong S^n$ , this inductively computes $H_k(S^n)$ .

Naturality

RemarkNaturality of the long exact sequence

The long exact sequence is natural: given a map of pairs $f : (X, A) \to (Y, B)$ , there is a commutative ladder connecting the two long exact sequences via the induced maps $f_*$ . This naturality is essential for functorial arguments and comparison theorems in algebraic topology.

The long exact sequence of a pair converts topological inclusion problems into algebraic exactness conditions, and its systematic application is the primary method for computing homology groups of CW complexes and manifolds.