$n$-connected $n$-dimensional CW complexes are contractible

Let \(X\) be an \(n\)-connected, \(n\)-dimensional CW complex. Thus \[ \pi_i(X)=0\qquad(0\le i\le n). \] Suppose some higher homotopy group were nonzero, and let \(m>n\) be the least integer with \[ \pi_m(X)\ne0. \] Then \(X\) is \((m-1)\)-connected. By the Hurewicz theorem, the Hurewicz map \[ \pi_m(X)\longrightarrow H_m(X;\mathbb Z) \] is an isomorphism. Hence \(H_m(X;\mathbb Z)\ne0\).

But \(X\) has no cells in dimensions greater than \(n\), so its cellular chain complex is zero above dimension \(n\). Therefore \[ H_m(X;\mathbb Z)=0\qquad(m>n), \] a contradiction. Thus every homotopy group of \(X\) is trivial.

The map \(X\to *\) is consequently an isomorphism on all homotopy groups. Since \(X\) and a point are CW complexes, Whitehead’s theorem implies that \(X\to *\) is a homotopy equivalence. Hence \[ \boxed{X\text{ is contractible}.} \]