$p$-local stable splitting of quasitoric manifolds

We show a homotopy decomposition of $p$-localized suspension $Sigma M_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $pi$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>dim M/2$. We also discuss non-triviality of $pi_{(p)}$ and $Sigma^inftypi$.