 Finite CW approximation from bounded homology

Let $X$ be simply connected, with finitely generated integral homology groups, and suppose $H_i(X;\mathbb{Z})=0$ for $i>n$, where $n\ge 2$. Then $X$ admits a weak homotopy equivalence $C\to X$ from a finite CW complex of dimension at most $n+1$. If $X$ already has CW complex homotopy type, this map is a homotopy equivalence.

Start with a finite $n$-dimensional CW complex $C_n$ and an $n$-connected map $f:C_n\to X$, constructed by the finite type CW approximation. This means that $f$ induces isomorphisms on homotopy groups below $n$ and a surjection in degree $n$; it does not mean that $C_n$ itself is $n$-connected.

The kernel of $H_n(C_n;\mathbb{Z})\to H_n(X;\mathbb{Z})$ is a finitely generated free abelian group, because $H_n(C_n;\mathbb{Z})$ is a subgroup of its finite free top cellular chain group. The Relative Hurewicz theorem represents a basis of this kernel by attaching $(n+1)$-cells whose maps extend into $X$. Their cellular boundary is injective, killing exactly that kernel without adding top homology. All integral homology groups now agree, so the homological Whitehead theorem proves the conclusion. The acyclic simply connected case is already weakly equivalent to a point.