Let \(G\) be a group and \(X\) a simply-connected space. Show that for the product \(K(G, 1) \times X\) the action of \(\pi_1\) on \(\pi_n\) is trivial for all \(n > 1\).
Solution
Let \(K=K(G,1)\) and let \[ p:\widetilde K\times X\longrightarrow K\times X \] be the universal covering. Since \(X\) is simply connected, the deck group is \(G\), acting by \[ g\cdot(u,x)=(gu,x). \] The projection \[ \widetilde K\times X\to X \] is a homotopy equivalence because \(\widetilde K\) is contractible. Under the induced identification \[ \pi_n(K\times X)\cong\pi_n(\widetilde K\times X)\cong\pi_n(X), \qquad n>1, \] every deck transformation acts as the identity on the \(X\)-factor. Hence it induces the identity on \(\pi_n\).
By the description of the \(\pi_1\)-action via deck transformations, \[ \boxed{\pi_1(K(G,1)\times X)\text{ acts trivially on }\pi_n(K(G,1)\times X),\ n>1.} \]