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 \[ j = ∑ j q : C ( R n , X ) → ∏ q ⩾ 0 Q D q ( R n , X ) j = \sum {{j_q}} :C({R^n},X) \to \prod \limits _{q \geqslant 0} {Q{D_q}({R^n},X)} \] is an E n {E_n} ring space and j j is a C n {\mathcal {C}_n} -map, where C n {\mathcal {C}_n} is the little n n -cubes operad. This implies that j j has an n n -fold delooping with domain Σ n X {\Sigma ^n}X . It also implies an algorithm for the calculation of j ∗ {j_{\ast }} and thus of each ( j q ) ∗ {({j_q})_{\ast }} on mod p \bmod \, p homology. When n = ∞ n = \infty and p = 2 p = 2 , this algorithm is the essential starting point for Kuhn’s proof of the Whitehead conjecture.