Theorem 11.7:

Suppose that $A$ is a deformation retract of $X$ via a map $r: X \longrightarrow A$. Then for $x_0\in A$, the maps $r_*:\pi_1(X, x_0)\longrightarrow \pi_1(A, x_0)$ and $i_*:\pi_1(A, x_0)\longrightarrow \pi_1(X, x_0)$ are isomorphisms.