On the Group Structure of [_S2,_Y]

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Let J(X) denote the James construction on a space X and Jn(X) be the nth stage of the James filtration of J(X). It is known that [J(X),Y] ∼=
lim←[Jn(X),Y] for any space Y . When X = S1, the circle, J(S1) = S1 = S2. Furthermore, there is a bijection between [J(S1),Y] and
the product ∞ i=2 πi(Y ), as sets. In this paper, we describe the group structure of [Jn(S1),Y] by determining the co-multiplication structure on the suspension Jn(S1).