Some Properties of Fox Torus Homotopy Groups

In 1945, R. Fox introduced the so-called Fox torus homotopy groups in which the usual homotopy groups are embedded and their Whitehead products are expressed as commutators. A modern treatment of Fox torus homotopy groups and their generalization were given by the authors in an earlier work. In this note, we further explore these groups and their properties. We discuss co-multiplications on Fox spaces and a Jacobi identity for the generalized Whitehead products. Introduction First, we recall from [2], [3] the definition of the n-th Fox torus homotopy group of a pointed space X, for n ≥ 1. Let x0 be a basepoint of X, then τn(X, x0) = π1(X T , x0) where X n−1 denotes the space of unbased maps from the (n − 1)torus T n−1 to X and x0 is the constant map at x0. When n = 1, τ1(X, x0) = π1(X, x0). To re-interpret Fox’s result, we showed in [5] that τn(X, x0) ∼= [Σ(T n−1 ⊔ ∗), X] Date: March 6, 2008. 2000 Mathematics Subject Classification. Primary: 55Q05, 55Q15, 55Q91; secondary: 55M20.