Theorem 29.6:

A short exact sequence of complexes (29.16) induces a long exact sequence in homology

$\begin{CD} @> >> H_n(L) @> H_n(f) >> H_n(G) @> H_n(g) >> H_n(K) @> {\delta_n}>> H_{n-1}(L) @> >> \\ \end{CD}$(29.17)

where the map $\delta_n:H_n(K)\longrightarrow H_{n-1}(L)$ known as the connecting homomorphism is given by the formula

$$\delta_n\overline{k_n} = \overline{f_{n-1}^{-1} \partial _n g_n^{-1}(k_n)},\quad k_n\in Z_n(K)\eqno(29.18)$$

Here $\overline{k_n}$ refers to the homology class of $k_n \in Z_n(K)$ and $g^{-1}(k_n)$ refers to any pre-image of $k_n$.