A Γ-Whitehead product for track groups and its dual

Abstract

In this paper, we generalize the product defined by Hardie and Jansen in [3]. The generalized product gives a pairing [·, ·] ^W _Γ : [ΓX ∧ W, Z] × [ΓY ∧ W, Z] → [Γ(X ∧ Y) ∧ W, Z]. When Γ = S ¹, X = S ^m−1 and Y = S ⁿ⁻¹, it is the product of Hardie and Jansen. We study properties of it such as bilinearity, Jacobi identity and some formulas in [4]. Furthermore, we consider the dual product and investigate various properties corresponding under the principle of Eckmann-Hilton duality.