The Double Suspension and Exponents of the Homotopy Groups of Spheres

Abstract

A simply connected space X is said to have exponent pk at the prime p if pk annihilates the p-primary component of the homotopy groups w*(X). Relations between the double suspension Y2: S2-'--> Q2S2'+ and the p-th James-Hopf invariant Hp: QS2+' -> QS2Pn+' give that S2f+ has p2f as an exponent at any prime p. This was proved for p = 2 by I. M. James [8] and for p odd by H. Toda [15]. Let p be an odd prime. M. Barratt [unpublished] conjectured that pf is an exponent for S2f+' at p. No smaller power of p could be an exponent since B. Gray [6] has shown that w*(S2f+') contains infinitely many elements with order pf. These elements stabilize mod p to elements in the image of the stable J-homomorphism. By studying a map derived from the p-th James-Hopf invariant, P. Selick [13] has shown that Barratt's conjecture is true when n = 1. Suppose p is greater than 3. In our previous paper [4], we showed that there exists a map w: Q2S2n+l -> S2,n-1 of spaces localized at p which is of degree p on the bottom cell. We used this map to show that pn+l is an exponent for S2f+ ' at p. In the present paper, we refine these methods. We show that w can be chosen so that the composition ?2w with the double suspension is the double looping of the degree p map. This enables us to apply induction to Selick's result and thus prove Barratt's conjecture for all n. The map w: Q2S2n+1_> S21'-1 may have other implications in homotopy theory. Let D(n) be its homotopy theoretic fibre and let C(n) be the homotopy theoretic fibre of the double suspension ?2: S2f1-1 -___ Q2S2n+1. Homological calculations and certain fibrations support the conjecture that C(n) has the homotopy type of the loop space QD(pn). (As a by-product of a new proof of Selick's theorem, we show that C(1) has the homotopy type of QD(p).)