Theorem (Hurewicz): Let $X$ be an $(n-1)$-connected CW complex with $n \geq 2$. Then $\tilde{H}_k(X) = 0$ for $k < n$ and $h : \pi_n(X) \to H_n(X)$ is an isomorphism.

Step 1: Reduction to a CW complex with trivial $(n-1)$-skeleton.

Since $X$ is $(n-1)$-connected, by cellular approximation and CW approximation, we may assume $X$ is a CW complex with $X^{(n-1)} = \{x_0\}$ (a single point). This is because $\pi_k(X) = 0$ for $k < n$ allows us to collapse the $(n-1)$-skeleton to a point without changing homotopy groups (by the Whitehead theorem applied to the quotient map).

With this CW structure, $X$ has no cells in dimensions $1$ through $n-1$, so the cellular chain complex has $C_k^{CW}(X) = 0$ for $1 \leq k \leq n-1$, giving $H_k(X) = 0$ for $1 \leq k < n$.

Step 2: Identification of $\pi_n(X)$ and $H_n(X)$ via the attaching maps.

The $n$-cells of $X$ are attached by maps $\varphi_\alpha : S^{n-1} \to X^{(n-1)} = \{x_0\}$, which are all constant. Therefore each $n$-cell gives a map $\Phi_\alpha : S^n \to X^{(n)} = \bigvee_\alpha S^n_\alpha$ (the wedge of $n$-spheres), representing an element $[\Phi_\alpha] \in \pi_n(X^{(n)})$.

For the wedge of spheres, we have: $\pi_n\left(\bigvee_\alpha S^n_\alpha\right) \cong \bigoplus_\alpha \mathbb{Z} \cong H_n\left(\bigvee_\alpha S^n_\alpha\right)$

The first isomorphism holds because the wedge of $n$-spheres is $(n-1)$-connected and the inclusion $S^n_\alpha \hookrightarrow \bigvee S^n_\alpha$ generates a free summand. The second is immediate from cellular homology.

Step 3: The Hurewicz map commutes with attaching.

Consider the exact sequence from the pair $(X, X^{(n)})$. Since $X$ is obtained from $X^{(n)}$ by attaching cells of dimension $> n$, and these higher-dimensional cells do not affect $\pi_n$ or $H_n$ (by cellular approximation and the cellular chain complex), we have: $\pi_n(X) \cong \pi_n(X^{(n)}) / N$ $H_n(X) \cong H_n(X^{(n)}) / B$ where $N$ and $B$ are the subgroups generated by the attaching maps of the $(n+1)$-cells.

Step 4: The subgroups $N$ and $B$ correspond under the Hurewicz map.

Each $(n+1)$-cell has attaching map $\psi_\beta : S^n \to X^{(n)}$. The element $[\psi_\beta] \in \pi_n(X^{(n)})$ maps to the cellular boundary $\partial_{n+1}(e^{n+1}_\beta) \in C_n^{CW}(X^{(n)})$ under the Hurewicz map. Since $h$ maps $N$ isomorphically onto $B$, it descends to an isomorphism on the quotients: $h : \pi_n(X) = \pi_n(X^{(n)}) / N \xrightarrow{\;\cong\;} H_n(X^{(n)}) / B = H_n(X)$

This completes the proof. $\square$