On the group structure of $[Ω\mathbb S^2, ΩY]$

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),ΩY]\cong \lim\limits_{\leftarrow} [J_n(X),ΩY]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=ΩΣ\mathbb S^1=Ω\mathbb S^2$. Furthermore, there is a bijection between $[J(\mathbb S^1),ΩY]$ and the product $\prod_{i=2}^\infty π_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n(\mathbb S^1),ΩY]$ by determining the co-multiplication structure on the suspension $ΣJ_n(\mathbb S^1)$.

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