CW complexes with prescribed homotopy groups

Let the prescribed fundamental group be \(G_1\). For \(i>1\), the prescribed group \(G_i\) is necessarily abelian, and suppose we are given a homomorphism \[ \rho_i:G_1\to\operatorname{Aut}(G_i) \] for each \(i>1\).

Choose functorial CW or simplicial models \(K(G_i,i)\). Functoriality lets each automorphism of \(G_i\) act on \(K(G_i,i)\), so \(\rho_i\) gives a \(G_1\)-action on this Eilenberg–MacLane space. Form \[ F=\prod_{i>1}K(G_i,i) \] with the diagonal \(G_1\)-action. Replace the product by the standard weak product CW model if desired; it has \[ \pi_i(F)\cong G_i\qquad(i>1), \quad \pi_1(F)=0. \]

Let \(EG_1\) be a contractible free \(G_1\)-CW complex and define the Borel construction \[ X=EG_1\times_{G_1}F. \] There is a fiber bundle \[ F\longrightarrow X\longrightarrow BG_1=K(G_1,1). \] Since \(F\) is simply connected, the long exact homotopy sequence gives \[ \pi_1(X)\cong G_1, \qquad \pi_i(X)\cong\pi_i(F)\cong G_i\quad(i>1). \] The universal cover is \(EG_1\times F\simeq F\), and a deck transformation \(g\in G_1\) acts on the \(i\)-th homotopy group exactly through the chosen action \(\rho_i(g)\). Hence \[ \boxed{X\text{ realizes all the prescribed homotopy groups and prescribed }\pi_1\text{-actions}.} \]