Higher homotopy commutativity and extension of maps

Abstract

Let X X denote the cartesian product of based spaces, X = X 1 × ⋯ × X n X = {X_1} \times \cdots \times {X_n} , and A = X 1 ∨ ⋯ ∨ X n A = {X_1} \vee \cdots \vee {X_n} , the subspace consisting of their one-point union. Further, let g : A → Y g:A \to Y be a map, for Y Y any based space. This article develops a criterion for the extendibility of g g to a map G : X → Y G:X \to Y . The criterion is in terms of higher products which live in the Pontryagin ring of Ω Y \Omega Y , the loop space of Y Y .