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


Authors: J. Caruso, F. R. Cohen, J. P. May and L. R. Taylor
Journal: Trans. Amer. Math. Soc. 281 (1984), 243-283
MSC: Primary 55P35; Secondary 18F25, 19L64, 55P47, 55Q05, 55Q25, 55S15
DOI: https://doi.org/10.1090/S0002-9947-1984-0719669-2
MathSciNet review: 719669
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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

$\displaystyle j = ({j_q}):{\Omega ^n}{\Sigma ^n}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{2nq}}{\Sigma ^{2nq}}{D_q}({R^n},X)$

is a ring space, and $ j$ is an exponential $ H$-map. There is a total Segal map

$\displaystyle s = \mathop \times \limits_{q \geqslant 0} \;{s_{q}}:\mathop \tim... ... \mathop \times \limits_{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}$

which is a ring map between ring spaces. There is a total partial power map

$\displaystyle k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}$

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

$\displaystyle \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigm... ...es \limits_{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.$

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^ + })$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0719669-2
Keywords: James maps, Kahn-Priddy theorem, iterated loop space, splitting theorem, coefficient system, group completion
Article copyright: © Copyright 1984 American Mathematical Society

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