Definition 37.1:

(i) Given a topological space $X$ and a subspace $A$, $S_n(A)$ may be regarded as a subgroup of $S_n(X)$ and the group $S_n(X, A)$ of relative $n-$chains is the quotient group $S_n(X)/S_n(A)$.

(ii) For each $n = 1, 2, \dots,$ we define the boundary maps ${\overline\partial }_n : S_n(X, A)\longrightarrow S_{n-1}(X, A)$ as

$${\overline\partial }_n \overline{c} = \overline{\partial _n c} \eqno(37.1)$$

It is readily verified that ${\overline\partial }_{n-1}\circ {\overline \partial }_n = 0$ leading to the quotient complex $S(X)/S(A)$ consisting of the sequence of groups $\{S_n(X, A)\}$ and the boundary maps (37.1).

(iii) The homology groups of the quotient complex $S(X)/S(A)$ are called the relative homology groups and are denoted by the symbol $H_n(X, A)$.

For a slightly more explicit description of these groups we introduce the group $Z_n(X, A)$ of relative $n-$cycles and the group $B_n(X, A)$ of relative boundaries. The group $Z_n(X, A)$ is the subgroup of $S_n(X)$ consisting of chains $c \in S_n(X)$ such that the boundary $\partial _nc$ is a chain in $A$. That is,

$$Z_n(X, A) = \{c\in S_n(X) / \partial _nc \in S_{n-1}(A) \}. \eqno(37.2)$$

In keeping with the convention that $S_{-1}(A) = \{0\}$ (see definition (29.5)), $Z_0(X, A) = S_0(X)$. We see that $c \in Z_n(X, A)$ if and only if $\overline{c}$ is in the kernel of ${\overline\partial }_n$. Likewise the group $B_n(X, A)$ of relative boundaries is defined to be the subgroup of $S_n(X)$ consisting of chains $c \in S_n(X)$ such that

$$c = \partial _{n+1}c^{\prime}\;$$mod$$(S_{n}(A)),$$

for some $c^{\prime}\in S_{n+1}(X)$. In other words there exists $c^{\prime}\in S_{n+1}(X)$ and $a \in S_n(A)$ such that

$$c - \partial _{n+1}c^{\prime} = a.$$

Obviously $c \in B_n(X, A)$ if and only if $\overline{c}$ belongs to the image of ${\overline\partial }_{n+1}$ whereby we conclude