Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups

Abstract

In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group $\tau = \left[ \sum\left(V \times WU*\right), X\right]$ in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of $\tau$, thereby improving a result of Varadarajan.