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 . In addition we will discuss other periodic phenomena. Our main results have been independently obtained by Bendersky using BP*. Our methods, however, are entirely geometric, and we actually construct the elements, rather than detect them. Our basic tool is the map .All our constructions are made in BΣp and transferred over.
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