Extended powers of spectra and a generalized Kahn-Priddy theorem
Let X be a connected infinite loop space. Then \(QX=\Omega^{\infty}\Sigma^{\infty}X\) splits as QX\(\simeq X\times X'\) where X' is the fibre of the structure map \(\theta\) : QX\(\to X\). Stably one has a splitting \(\bigvee f_ n: \bigvee_{n}\Sigma^{\infty}D_ nX\to \Sigma^{\infty}QX\) where \(D_ nX=E\Sigma^+_ n\wedge_{\Sigma_ n} X^{\wedge n}\) is the n-th extended power construction. The author shows (Theorem 1.1) that after localization this splitting induces retractions \((QD_ 2X)_{(2)}\to X'\!_{(2)}\), and \(Q(D_ 2X\vee D_ pX)_{(p)}\to X'\!_{(p)}\) for \(p>2\). For \(X=S^ 1\) this specializes to a delooped Kahn-Priddy theorem. The proof of Theorem 1.1 is quite short and noncomputational, using manipulations of the adjoint pair \((\Sigma^{\infty},\Omega^{\infty})\).
scientific article; zbMATH DE number 3905452