Relative singular homology

Let $B\subseteq X$ be a subspace of a topological space $X$ and let $A$ be an abelian group. The inclusion induces a subcomplex $C_\bullet(B;A)\subseteq C_\bullet(X;A)$ . The relative singular chain complex is

C_\bullet(X,B;A):=C_\bullet(X;A)/C_\bullet(B;A),

and the relative singular homology group is

H_n(X,B;A):=H_n(C_\bullet(X,B;A)).

Thus a relative cycle is a chain in $X$ whose boundary lies in $B$ , modulo chains in $B$ and relative boundaries.

Functoriality and exact sequence

A map of pairs $f:(X,B)\to(Y,D)$ induces maps on relative homology. The short exact sequence of chain complexes

0\to C_\bullet(B;A)\to C_\bullet(X;A)\to C_\bullet(X,B;A)\to0

gives the long exact sequence of the pair, relating $H_n(B;A)$ , $H_n(X;A)$ , and $H_n(X,B;A)$ .

Reference

See Allen Hatcher, Algebraic Topology, Chapter 2, author-hosted book.