Relative homology

Remarks

A class in $H_n(X, A)$ is represented by a relative cycle, a chain $c \in C_n(X)$ with $\partial c \in C_{n-1}(A)$. Two relative cycles give the same class when their difference is $\partial b + a$ with $b \in C_{n+1}(X)$ and $a \in C_n(A)$. For $A = \emptyset$, the group $H_n(X, \emptyset)$ is the singular homology $H_n(X)$. A map of pairs $f \colon (X, A) \to (Y, B)$, that is, a continuous map with $f(A) \subseteq B$, induces maps $f_* \colon H_n(X, A) \to H_n(Y, B)$.

Example: for $n \ge 1$, $H_n(D^n, S^{n-1}) \cong \Z$ and the other relative groups vanish. For $n = 1$, the path $\sigma(t) = 2t - 1$ in $D^1 = [-1, 1]$ is a relative cycle, since $\partial \sigma = (1) - (-1)$ lies in $C_0(S^0)$.

Non-example: in $X = [-1, 1]$ with $A = \set{-1}$, the same path is not a relative cycle, because its boundary contains the point $1 \notin A$.