Theorem 31.3:

Suppose $r: X \longrightarrow A$ is a retraction, then for every $n = 0, 1, 2,\dots$

$$H_n(r):H_n(X) \longrightarrow H_n(A)$$

is surjective and

$$H_n(j):H_n(A) \longrightarrow H_n(X)$$

is injective, where $j:A\longrightarrow X$ is the inclusion map.