James maps, Segal maps, and the Kahn-Priddy theorem

Abstract:

The standard combinatorial approximation $C({R^n},X)$ to ${\Omega ^n}{\Sigma ^n}X$ is a filtered space with easily understood filtration quotients ${D_q}({R^n},X)$. Stably, $C({R^n},X)$ splits as the wedge of the ${D_q}({R^n},X)$. 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 \[ j = (j_q): \Omega ^n \Sigma ^n X \bigtimes _{q \geqslant 0} \Omega ^{2nq} \Sigma ^{2nq} D_q(R^n, X) \] is a ring space, and $j$ is an exponential $H$-map. There is a total Segal map \[ s = \bigtimes _{q \geqslant 0} \;{s_{q}}:\bigtimes _{q \geqslant 0} \;{\Omega ^{2nq}}\,{\Sigma ^{2nq}}{D_q}({R^{n}},X)\; \bigtimes _{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}\] which is a ring map between ring spaces. There is a total partial power map \[ k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}\] which is an exponential $H$-map. There is a noncommutative binomial theorem for the computation of the smash power ${\Omega ^n}{\Sigma ^n}X \to {\Omega ^{nq}}{\Sigma ^{nq}}{X^{[q]}}$ in terms of the ${k_m}$ for $m \leqslant q$. The composite of $s$ and $j$ agrees with the composite of $k$ and the natural inclusion \[ \bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}} \to \bigtimes _{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.\] This analysis applies to essentially arbitrary spaces $X$. When specialized to $X = {S^0}$, it implies an unstable version of the Kahn-Priddy theorem. The exponential property of the James maps leads to an analysis of the behavior of loop addition with respect to the stable splitting of ${\Omega ^n}{\Sigma ^n}X$ when $X$ is connected, and there is an analogous analysis relating loop addition to the stable splitting of $Q({X^ + })$.