Theorem 26.8:

Let $X\sqcup_f E^k$ be the space obtained by attaching a $k$ cell to a path connected space $X$ via a map $f:S^{k-1}\longrightarrow X$. Then for any choice of base point in $f(S^{k-1})$,

(i)

$\pi_1(X\sqcup_f E^k, x_0) = \pi_1(X, x_0)$ if $k\geq 3$.

(i)

$\pi_1(X\sqcup_f E^2, x_0) = \pi_1(X, x_0)/\langle$   im$\;f_*\rangle$.