CW approximation

A CW approximation of a space $X$ is a CW complex $X_{\mathrm{cw}}$ with a weak homotopy equivalence $X_{\mathrm{cw}}\to X$. The realization of the singular simplicial set of $X$ supplies one functorially.

For simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

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.