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).

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