A note on the Kahn-Priddy map (Q1098454)

Let \(L^ k_ p\) be the k-skeleton of the infinite dimensional lens space mod a prime p. The main purpose of this paper is to determine the orders \(\lambda_{2n}\) and \(\lambda '_{2n}\), where \(\lambda_{2n}\in \pi^ 0(L_ p^{2n})\) is a Kahn-Priddy map and \(\lambda '_{2n}=\lambda_{2n}| L_ p^{2n-1}\). Our method is to use the d- or e-invariant argument and to use the suspension order of \(L_ p^{2n}/L_ p^{2p-4}\). For an odd prime p, our result coincides with Nishida's. As a byproduct of our proof, the periodic family of elements of \(\pi_{2n-1}(S^ 0)\) are recovered. In the appendix Toda's result about the stable order of \(L_ 2^{2n}\) is shortly proved.