On Fox spaces and Jacobi identities

TL;DR

This paper explores Fox spaces and their properties, focusing on co-multiplications and Jacobi identities for generalized Whitehead products, extending classical homotopy group concepts introduced by Fox in 1945.

Contribution

It provides a modern analysis of Fox torus homotopy groups, introducing new properties and identities for generalized Whitehead products and co-multiplications.

Findings

Establishment of co-multiplications on Fox spaces

Derivation of a Jacobi identity for generalized Whitehead products

Extension of Fox's original homotopy group framework

Abstract

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 has been given and studied. 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 and the $\Gamma$-Whitehead products.