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

TL;DR

This paper investigates the algebraic group structure of homotopy classes of maps from James construction stages on the circle to loop spaces, revealing the co-multiplication structure on their suspensions.

Contribution

It characterizes the group structure of $[J_n(S^1), \, \Omega Y]$ by analyzing the co-multiplication on the suspension of James construction stages.

Findings

Determined the group structure of $[J_n(S^1), \Omega Y]$

Described the co-multiplication on $\Sigma J_n(S^1)$

Connected the James construction to homotopy group products.

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 $[J(X),\Omega Y]\cong \lim\limits_{\leftarrow} [J_n(X),\Omega Y]$ 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 $\Sigma J_n(\mathbb S^1)$.