Homotopy-group effect of attaching higher cells (${\pi }_i(X{\cup }_{\alpha }{e}^{i+1})={\pi }_i(X)/⟨{\pi }_1(X)\cdot [\alpha ]⟩(i\ge 2)$)

Attaching cells of dimension $i+1$ to a connected CW complex, for $i\ge 2$, preserves all homotopy groups below degree $i$. In degree $i$ it takes the quotient by the subgroup generated by the attaching maps and their translates under the fundamental group action. This follows from the relative homotopy sequence and the generation of the first nonzero relative group by the new cells. In particular, a single attaching map representing a generator of ${\pi }_i\cong \mathbb{Z}$ kills that group even when the fundamental group is nontrivial. Higher attachments can create higher homotopy groups, which may then be killed at later stages.