James Maps, Segal Maps, and the Kahn-Priddy Theorem

Abstract

The standard combinatorial approximation <!-- MATH $C({R^n},X)$ --> to <!-- MATH ${\Omega ^n}{\Sigma ^n}X$ --> is a filtered space with easily understood filtration quotients <!-- MATH ${D_q}({R^n},X)$ --> . Stably, <!-- MATH $C({R^n},X)$ --> splits as the wedge of the <!-- MATH ${D_q}({R^n},X)$ --> . We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map <!-- MATH \begin{displaymath} j = ({j_q}):{\Omega ^n}{\Sigma ^n}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{2nq}}{\Sigma ^{2nq}}{D_q}({R^n},X) \end{displaymath} -->