Splitting of certain spaces CX

There are a number of theorems to the effect that spaces of the form Q. n 11 n X split stably into wedges of simpler spaces when X is connected (by which we mean pathwise connected). The proofs generally proceed by exploitation of combinatorially manageable approximations for Q. n *L n X. The earliest theorem of this kind is due to Milnor(18), who exploited the semi-simplicial version of the James construction on X to split ZQ ~LX into V 2>X lq \ where X [q) denotes the g-fold smash product of X with itself. Q>1 More recently, Kahn(iO) proved that the suspension spectrum of QX = lim £2 7l £ n X splits as the wedge of the suspension spectra of the extended powers ~* D q X = E-Lt Ax q XM, where S g is the symmetric group on q letters, E"L q is a contractible space on which S g acts freely, and Y+ denotes the union of a space Y and a disjoint basepoint. He exploited Barratt's semisimplicial approximation to QX, and a proof of this splitting has also been given by Barratt and Eccles(i). A bit later, Snaith(22) proved that the suspension spectrum of Q m 2 n X for 1 ^ n splits as the wedge of the suspension spectra of the extended powers where c ^ n q is Boardman and Vogt's space of g-tuples of little «-cubes disjointly embedded in R n. He exploited May's approximation C n X to D. n "L n X. C n X comes with a filtration, and Snaith also proved that 'L t F T C n X splits as the wedge of the 2*Z) n q X for 1 < q ^ r and a certain t depending on r and n. Snaith's stable splittings were later rederived, in the general context of operads, by Reedy (20). These splittings have hada number of applications inhomotopy theory. In particular, Mahowald (12) has recently made striking use of very special cases of Snaith's splittings. The proofs of these splittings generally involve rather complicated combinatorial arguments. It is one purpose of the present paper to give transparently elementary proofs of all these results (modulo the approximation theorem for D ra 2 n X). The simplicity of our construction of the requisite splitting maps makes these decomposi-tions considerably easier to work with. For example, the first author has obtained …