Theorem 23.5.
Let \(X\) be a \(1\)-connected CW complex. Then \(X\) has homology of finite type if and only if \(X\) has homotopy of finite type.
Section 23.4 Finite Type: Homology vs. Homotopy
Proof.
(\(\Leftarrow\)) Homotopy of finite type \(\Rightarrow\) homology of finite type. Use the Postnikov tower: we show by induction on \(n\) that each \(X_{\leq n}\) has homology of finite type. The bottom space \(X_{\leq 1} \simeq *\) is fine. Suppose \(X_{\leq n-1}\) has homology of finite type. The map \(X_{\leq n} \to X_{\leq n-1}\) is a fibration with fibre \(K(\pi_n X, n)\text{;}\) by Proposition 23.4 (applied to the finitely generated group \(\pi_n X\)) and Theorem 23.2, \(X_{\leq n}\) has homology of finite type. Realising \(X\) as the inverse limit of the Postnikov tower and using that \(X \to X_{\leq n}\) induces isomorphisms on \(H_i\) for \(i \leq n - 1\) (an application of relative Hurewicz to the pair \((X_{\leq n}, X)\text{,}\) which is \(n\)-connected), we conclude \(H_i(X)\) is finitely generated for each \(i\text{.}\)
(\(\Rightarrow\)) Homology of finite type \(\Rightarrow\) homotopy of finite type. Use the Whitehead tower of \(X\text{.}\) Inductively, suppose the \((n-1)\)-connected cover \(X_{>(n-1)}\) has homology of finite type. The Hurewicz theorem applied to \(X_{>(n-1)}\) gives
\begin{equation*}
\pi_n(X) \;\cong\; \pi_n(X_{>(n-1)}) \;\cong\; H_n(X_{>(n-1)}),
\end{equation*}
which is finitely generated. Hence \(K(\pi_n X, n)\) has homology of finite type by Proposition 23.4. Applying Theorem 23.2 to the fibration \(X_{>n} \to X_{>(n-1)} \to K(\pi_n X, n)\) shows \(X_{>n}\) has homology of finite type, completing the inductive step.
