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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.

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