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James maps and $ E\sb{n}$ ring spaces


Authors: F. R. Cohen, J. P. May and L. R. Taylor
Journal: Trans. Amer. Math. Soc. 281 (1984), 285-295
MSC: Primary 55P35; Secondary 55P47, 55Q25, 55S12
DOI: https://doi.org/10.1090/S0002-9947-1984-0719670-9
MathSciNet review: 719670
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Abstract: We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James map

$\displaystyle j = \sum {{j_q}} :C({R^n},X) \to \prod\limits_{q \geqslant 0} {Q{D_q}({R^n},X)} $

is an $ {E_n}$ ring space and $ j$ is a $ {\mathcal{C}_n}$-map, where $ {\mathcal{C}_n}$ is the little $ n$-cubes operad. This implies that $ j$ has an $ n$-fold delooping with domain $ {\Sigma^n}X$. It also implies an algorithm for the calculation of $ {j_{\ast}}$ and thus of each $ {({j_q})_{\ast}}$ on $ \bmod\, p$ homology. When $ n = \infty$ and $ p = 2$, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0719670-9
Keywords: James maps, $ {E_n}$ ring space, operad, coefficient system, homology operations
Article copyright: © Copyright 1984 American Mathematical Society

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