James maps and $E_{n}$ ring spaces

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 \[ j = \sum {{j_q}} :C({R^n},X) \to \prod \limits _{q \geqslant 0} {Q{D_q}({R^n},X)} \] 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.