Algebraic topology Homology and cohomology Theorem

Long exact sequence of a pair

Builds on Relative homology Singular homology Exact sequence Reduced homology

Statement

Let $X$ be a topological space and let $A \subseteq X$ be a subspace. Let $i \colon A \to X$ be the inclusion and $j \colon (X, \emptyset) \to (X, A)$ the identity map seen as a map of pairs. For $n \ge 1$ let $\partial \colon H_n(X, A) \to H_{n-1}(A)$ send the class of a relative cycle $c$ to the class of the cycle $\partial c$ in $A$ . Then the sequence of singular homology and relative homology groups $\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \xrightarrow{i_*} \cdots \to H_0(X) \xrightarrow{j_*} H_0(X, A) \to 0$ is exact. It is natural with respect to maps of pairs.

Remarks

The chain complexes form a short exact sequence $0 \to C_\bullet(A) \to C_\bullet(X) \to C_\bullet(X, A) \to 0$ . The long exact sequence in homology of a short exact sequence of chain complexes gives the result, and its connecting map is the $\partial$ above. If $A$ is nonempty, the same sequence holds with reduced homology $\tilde H_n(A)$ and $\tilde H_n(X)$ in place of $H_n(A)$ and $H_n(X)$ .

Example: for $(D^n, S^{n-1})$ with $n \ge 1$ , the disc has $\tilde H_k(D^n) = 0$ for all $k$ . Exactness of the reduced sequence then forces $\partial \colon H_k(D^n, S^{n-1}) \to \tilde H_{k-1}(S^{n-1})$ to be an isomorphism for every $k$ .

Going deeper

For a student

This is the exactness axiom of the Eilenberg–Steenrod axioms. Combined with excision, it computes the homology of spheres by induction on dimension, and it gives the long exact sequences behind cellular homology. The same algebra gives a long exact sequence for a triple $B \subseteq A \subseteq X$ and, dually, for singular cohomology with the arrows reversed.

Reading path

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Tag	`AX4ZM9BV3Z`
URI	`m2://AX4ZM9BV3Z`
Kind	statement, theorem
Writing	W0 stub
MSC	55N10
Status	This copy of the site has no build data. Run `m2 status AX4ZM9BV3Z`.
About	Relative homology
Uses	Relative homology, Singular homology, Exact sequence, Reduced homology
Changed	2 November 2026