Theorem 37.6:

A retraction $r: X \longrightarrow A$ gives for each $n = 0, 1, 2,\dots$ a short exact sequence

$$\begin{CD} 0@> >> H_n(A) @> H_n(i) >> H_n(X) @> H_n(p) >> H_n(X, A) @> >> 0. \end{CD} \eqno(37.7)$$

in Each of these short exact sequences splits. Thus

$$H_n(X) = H_n(A) \oplus H_n(X, A),\quad n = 0, 1, 2,\dots$$