James Maps and E n Ring Spaces

Abstract

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 mapis an En ring space and) is a en-map, where en is the little n-cubes operad. This implies that) has an n-fold delooping with domain~n X. It also implies an algorithm for the calculation of) * and thus of each ()q)* on mod p homology. When n == 00 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 lames maps i q : ex~QDq(e, X) used in [4] to obtain the stable splittings of spaces ex. In particular, we proved that if the given coefficient system e has a sum, then the product of the targets is a ring space and the product of the iq's is an exponential H-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 e is a "module" over an operad § and we deduce, essentially, thatq~O is a §-map. Here IIq~o 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 q~O D q ( e, X)) 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][10][11][12][13].When eis e(R n ), our results imply that (jq) has an n-fold delooping }:nX~Bn( IT QDq(W,X)).q~l

Stable Homotopy Around the Arf-Kervaire Invariant

“…For example, [62] was the basis of Nick Kuhn's proof of the Whitehead conjecture at odd primes -the analogue of the 2-adic version proved by Dan Kahn and Stewart Priddy (see § 1.5.1). For example, [62] was the basis of Nick Kuhn's proof of the Whitehead conjecture at odd primes -the analogue of the 2-adic version proved by Dan Kahn and Stewart Priddy (see § 1.5.1).…”

Homotopy Methods in Algebraic Topology

“…Many of Peter's significant papers do not specifically fit into any of the above categories, for example the joint work with Zabrodsky on H* Spin( n), with several collaborators on stable splittings [26], [27], [46], [48], [49], on fibrewise localization and completion [31], on realization of Eilenberg-MacLane spaces as Thorn spectra [33], a note on the Bockstein and Adams spectral sequences [34], on the dual Whitehead theorems [4 7], on characteristic classes in Borel cohomology [56], on a generalization of Smith theory [57] a note on weak equivalences and quasifibrations [67], and many others.…”