Lemma 9.3:

Suppose $r: X \longrightarrow A$ is a retraction, $j:A\longrightarrow X$ is the inclusion, then for $a \in A$

$$r_{\ast}:\pi_1(X,a) \longrightarrow \pi_1(A,a)$$

is surjective and

$$j_{\ast} : \pi_1(A,a) \longrightarrow \pi_1(X,a)$$

is injective.