Relative Homology Groups

Given a space $X$ and a subspace $A\subset X$ , define $C_n(X,A):=C_n(X)/C_n(A)$ . Since the boundary map $\partial: C_n(X)\to C_{n-1}(X)$ takes $C_n(A)$ to $C_{n-1}(A)$ , it induces a quotient boundary map $\partial: C_n(X,A)\to C_{n-1}(X,A)$ .

We have a chain complex $\displaystyle \dots\to C_{n+1}(X,A)\xrightarrow{\partial_{n+1}}C_n(X,A)\xrightarrow{\partial_n}C_{n-1}(X,A)\to\dots$ where $\partial^2=0$ holds. The relative homology groups $H_n(X,A)$ are the homology groups $\text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1}$ of this chain complex.

Relative cycles
Elements of $H_n(X,A)$ are represented by relative cycles: $n$ – chains $\alpha\in C_n(X)$ such that $\partial\alpha\in C_{n-1}(A)$ .

Relative boundary
A relative cycle $\alpha$ is trivial in $H_n(X,A)$ iff it is a relative boundary: $\alpha=\partial\beta+\gamma$ for some $\beta\in C_{n+1}(X)$ and $\gamma\in C_n(A)$ .

Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
$\begin{aligned} \dots\to H_n(A)\xrightarrow{i_*}H_n(X)\xrightarrow{j_*}H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)&\xrightarrow{i_*}H_{n-1}(X)\to\dots\\ &\dots\to H_0(X,A)\to 0. \end{aligned}$

The boundary map $\partial:H_n(X,A)\to H_{n-1}(A)$ is as follows: If a class $[\alpha]\in H_n(X,A)$ is represented by a relative cycle $\alpha$ , then $\partial[\alpha]$ is the class of the cycle $\partial\alpha$ in $H_{n-1}(A)$ .