Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S~(2n-1) -> T_(2n-1) -> (OMEGA)S~(2n+1) whose connecting map is degree p~(r). In a long and complex monograph, Anick constructed such a fibration for p >= 5 and r >= 1. Using new methods we give a much more conceptual construction which is also valid for p velence 3 and r >= 1. We go on to establish an H space structure on T_(2n-1) and use this to construct a secondary EHP sequence for the Moore space spectrum.
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