Math. Proc. Camb. Phil. Soc. (1984), 96, 95 Printed in Great Britain 95 Unstable families related to the image of J BY BRAYTON GRAY University of Illinois at Chicago (Received 12 July 1983; revised 23 January 1984) The object of this paper is to describe certain families of unstable elements in the homotopy groups of spheres at an odd prime. In so doing we completely account for the image of J as possible Hopf invariants of unstable elements. The analogous result for p = 2 was obtained in [13]. In addition we will discuss other periodic phenomena. Our main results have been independently obtained by Bendersky[5] using BP*. Our methods, however, are entirely geometric, and we actually construct the elements, A rather than detect them. Our basic tool is the map i?2 p ->• Q(8°). AH our constructions are made in BTP and transferred over. We begin by describing the behaviour. Let q = 2(p — 1). and v = v p (n). Then Im J at the prime p lies entirely in the nq— 1 stem and has order p v+1 . To describe this unstably we look at a typical example. Suppose v = 2. Sphere nq— 1 stem S 3 Z v W 8 2r+1 , r $s 3 Suspension is a monomorphism. This pattern persists in its simplicity and was first proven in [8]. An alternative proof is presented here (Proposition 13). The unstable families lie in the nq — 2 stem. With v = 2 we have Sphere nq — 2 stem S 3 S* S 2 r+1 n—3 Z P The first two suspensions are monomorphisms and the last two are epimorphisms. In between, suspension is multiplication by p. The persistence of this pattern is presented as Theorems 12 and 14 here. These verify conjectures 3-2-13 and 3-2-16 of [7]. The last group is 7r m -2+2n-i(S 2n ~ 1 ) an( i *hi s i s ^ ne fa" 8 *homotopy group of/S 2 ™" 1 that is not stable. The kernel of suspension is cyclic of order p. This is analogous to the Whitehead product [t, i] e ^ 4w _ 3 ('S' 2re ~ 1 ) for p = 2. We label this element w n and call it the Whitehead element. It is a consequence of our constructions that w n is a 2^-fold suspension but not a 2v + 2-fold suspension. This is a mod^j version of the vector field problem. We also study other periodic phenomena and prove conjecture 3-2-17 of [7] (see Propositions 17-20 and 23). 1. In this section we will discuss reducibility and coreducibility for truncated lens spaces. Let L = £°°/Z p and L n and ^-skeleton of L. Thus L 2n+1 = S 2n+1 /Z p is an orientable manifold. Write L% = L a jL b - x .