= Relative homology connecting homomorphism
{title2=$\delta:H_n(B,A)\to H_{n-1}(A)$}
A relative cycle is represented by a <singular chain> $b$ whose boundary lies in $A$. Define $\delta[\bar b]=[\partial b]$. If a representative changes by $\partial c+a$ with $a\in C_n(A)$, its boundary changes by $\partial a$, an $A$-boundary. Thus the map is well-defined and additive. Its image is exactly the kernel of $H_{n-1}(A)\to H_{n-1}(B)$: a cycle in $A$ dies in $B$ precisely when it is the boundary of a relative cycle lifted to $B$. This proves the corresponding exactness in the <long exact sequence in relative homology> directly.
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