James maps and 𝐸_{𝑛} ring spaces

Cohen, F. R.; May, J. P.; Taylor, L. R.

doi:10.1090/S0002-9947-1984-0719670-9

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

is a ${\mathcal{C}_n}$ -map, where ${\mathcal{C}_n}$ is the little

-cubes operad. This implies that

has an

-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

, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture.

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