$\pi_k$ of a CW complex is already determined by its $(k+1)$-skeleton.Suggest a slogan
Let \(X\) be a CW complex and \(k\geq 1\). The inclusion \(X^{k+1}\injects X\) of the \((k+1)\)-skeleton induces an isomorphism \(\pi_k(X^{k+1})\cong\pi_k(X)\) [@Hat02]. In particular, \(\pi_1(X)\) depends only on the \(2\)-skeleton [@Hat02], and attaching cells of dimension \(k+2\) or higher does not change \(\pi_k\).