On the group structure of $[\Omega {\mathbb{S}}^2,\Omega Y]$

Abstract

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 $[J(X),\Omega Y]\cong \underleftarrow{\lim }[J_n(X),\Omega Y]$ for any space $Y$. When $X={\mathbb{S}}^1$, the circle, $J({\mathbb{S}}^1)=\Omega \Sigma {\mathbb{S}}^1=\Omega {\mathbb{S}}^2$. Furthermore, there is a bijection between $[J({\mathbb{S}}^1),\Omega Y]$ and the product ${\prod }_{i=2}^{\infty }{\pi }_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n({\mathbb{S}}^1),\Omega Y]$ by determining the co-multiplication structure on the suspension $\Sigma J_n({\mathbb{S}}^1)$.

23 pages