Given a sequence of CW complexes \(K(G_n, n)\), \(n = 1, 2, \ldots\), let \(X_n\) be the CW complex formed by the product of the first \(n\) of these \(K(G_n, n)\)’s. Via the inclusions \(X_{n-1} \subset X_n\) coming from regarding \(X_{n-1}\) as the subcomplex of \(X_n\) with \(n\)th coordinate equal to a basepoint 0-cell of \(K(G_n, n)\), we can then form the union of all the \(X_n\)’s, a CW complex \(X\). Show \(\pi_n(X) \approx G_n\) for all \(n\).
Solution
Write \[ X_N=\prod_{i=1}^N K(G_i,i), \qquad X=\bigcup_N X_N. \] Each \(X_N\) is a subcomplex of \(X_{N+1}\). A map from a compact sphere into the CW complex \(X\) has image in a finite subcomplex, hence in some \(X_N\). The same is true for homotopies. Therefore \[ \pi_n(X)\cong \varinjlim_N\pi_n(X_N). \] For \(N\ge n\), the product formula gives \[ \pi_n(X_N) \cong \prod_{i=1}^N\pi_n(K(G_i,i)) \cong G_n, \] since \(K(G_i,i)\) has only one nonzero homotopy group, in degree \(i\). The bonding maps are the identity on this \(G_n\)-summand. Hence the direct limit stabilizes and \[ \boxed{\pi_n(X)\cong G_n\quad\text{for every }n\ge1.} \]