The value of 𝐽 at a Samelson product

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Introduction. At the Seattle Conference, 1963, M. G. Barratt asked the question, Can anything general be said about J: wk(SOn) -+7rk+n(Sn) on Samelson products? Lashof records [4] that Stasheff pointed out an answer to this question on the basis of results obtained by B. Steer in his Oxford thesis [8]. We offer here another answer, phrased in quite different terms, which is simple in both statement and proof. Let p: On+l--S be the usual bundle map with fiber On; in [5] we define the p-product: xr(O,) 0r8n,(Sn)-->ir,+8(Sn). The p-product of aEEirq(On) with O3=r8,(Sn) is denoted [a, ]3,. This product generalizes the J homomorphism: if Ln denotes the class of the identity map on Sn then J(a) [a, tn ], [5, Corollary 5.7].