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Let ρi ∈ πn+8i−1(S) denote an element which suspends to a generator of the image of the stable Jhomomorphism. We determine the image of the composite ρj ◦ ρk in v1-periodic homotopy v−1 1 πn+8i+8j−2(S ). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson.

Compositions in the ν 1 -periodic homotopy groups of spheres