On the higher Whitehead product

Thiago de Melo,Marek Golasiński

doi:10.1007/s40062-016-0153-z

Abstract

Porter’s approach is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie. Computations concerning the higher order Whitehead product for spheres and projective spaces are presented as well.

Highlights

Whitehead products play an important role in algebraic topology and its applications
The main result from [8] deals with the r th order spherical Whitehead product [ f1, . . . , fr ] for maps fi : Smi → X with r ≥ 3 defined in [9], and in particular, it states that the triple product [ f1, f2, f3] is a coset of a subgroup of πm−1(X ), where m = m1 + m2 + m3
Many properties which hold for the classical Whitehead product still hold for the triple one as well

Summary

Introduction

Whitehead products play an important role in algebraic topology and its applications. The main result from [8] deals with the r th order spherical Whitehead product [ f1, . Porter’s approach [15] was to construct the Whitehead map ωr for more than two spaces (see Eq (5)). They first appeared in the late 1960’s as a part of a research theme studying higher products, or higher structures, in homotopy theory. Higher Whitehead products have re-emerged as key players in the homotopy theory of polyhedral products. These are important objects in toric topology and are being increasingly used in geometric group theory and graph theory.

Preliminaries

Relative generalized Whitehead product

Generalized separation element

Higher order generalized Whitehead product

Triple spherical Whitehead products

Main result Theorem 1 provides the following commutative diagram

Some computations for spheres and projective spaces