= Homotopy-group effect of attaching higher cells
{title2=$\pi_i(X\cup_\alpha e^{i+1})=\pi_i(X)/\langle\pi_1(X)\cdot[\alpha]\rangle\quad(i\geq2)$}
Attaching cells of dimension $i+1$ to a connected <CW complex>, for $i\geq2$, 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.
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