James Maps and £ „ Ring Spaces

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=2jqC(R",X)-^\[QDq(R",X) <7>0 is an En ring space and j is a S„-map, where 6„ is the little n-cubes operad. This implies thaty has an n-fold delooping with domain 2"X It also implies an algorithm for the calculation of j and thus of each (jq) on mod p homology. When « = oo and p = 2, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture. In [2, §4], we analyzed the behavior with respect to pairings of the James maps j : CX -> QDq(G, X) used in [4] to obtain the stable splittings of spaces CA". In particular, we proved that if the given coefficient system G has a sum, then the product of the targets is a ring space and the product of the/ 's is an exponential 77-map. In fact, this is only a fragment of the full structure present in the most interesting examples. In the first two sections, we give a parametrized elaboration of this analysis. We assume the given coefficient system G is a "module" over an operad § and we deduce, essentially, that iJq):CX^][QDq(G,X) q»0 is a S-map. Here Uq>0 denotes the weak infinite product (all but finitely many coordinates of each point are the basepoint). Technically, we shall replace the target by its equivalent Q( V i>0 Dq(G, A")) and § by an equivalent, but larger, operad. This analysis requires the introduction of general notions of additive and multiplicative actions of operads on coefficient systems, and these generalizations of the definitions of an operad and of an action of one operad on another give considerable added flexibility to the theory of iterated loop spaces developed in [9-13]. When G is G(R"), our results imply that (jq) has an Ai-fold delooping 2"A"-£"( II QDq(R",X)\. \ q»\ ' Received by the editors January 20, 1983. 1980 Mathematics Subject Classification. Primary 55P35, 55P47, 55S12; Secondary 55Q25.