Homotopy equivalent skeleta

Let \(X\) and \(Y\) be homotopy equivalent CW complexes, and suppose neither has cells of dimension \(n+1\). Choose homotopy inverse maps \[ f:X\to Y, \qquad g:Y\to X. \] By cellular approximation, replace \(f\) and \(g\) by cellular maps. Then \[ f(X^n)\subset Y^n, \qquad g(Y^n)\subset X^n, \] so they restrict to maps \[ f_n:X^n\to Y^n, \qquad g_n:Y^n\to X^n. \]

We have homotopies \[ gf\simeq\operatorname{id}_X, \qquad fg\simeq\operatorname{id}_Y. \] Restrict the first to \(X^n\times I\). By cellular approximation for homotopies, relative to the two ends, it is homotopic to a cellular map \[ X^n\times I\longrightarrow X. \] The product CW structure on \(X^n\times I\) has dimension at most \(n+1\), so a cellular homotopy has image in \(X^{n+1}\). Since \(X\) has no \((n+1)\)-cells, \[ X^{n+1}=X^n. \] Hence \[ g_nf_n\simeq\operatorname{id}_{X^n} \] through maps into \(X^n\). The same argument gives \[ f_ng_n\simeq\operatorname{id}_{Y^n}. \] Therefore \(f_n\) and \(g_n\) are homotopy inverses, and \[ \boxed{X^n\simeq Y^n.} \]