Higher homotopy commutativity and extension of maps

Let $X$ denote the cartesian product of based spaces, $X = {X_1} \times \cdots \times {X_n}$, and $A = {X_1} \vee \cdots \vee {X_n}$, the subspace consisting of their one-point union. Further, let $g:A \to Y$ be a map, for $Y$ any based space. This article develops a criterion for the extendibility of $g$ to a map $G:X \to Y$. The criterion is in terms of higher products which live in the Pontryagin ring of $\Omega Y$, the loop space of $Y$.