ON CERTAIN THEOREMS OF G. W. WHITEHEAD

Abstract

This chapter presents the theorem of homotopy groups of spheres and rotation groups, and generalization of the Hopf invariant. It also discusses the operations on homotopy groups, the fundamental the operation, ×, and the composition operator, •, in relative homotopy groups. These are defined geometrically, but without reference to orientation. The product, join, suspension, and the composition, °, for absolute homotopy groups, are defined in terms of ×, and a given boundary operator ∂. A Hurewicz operator, or natural transformation, h, from homotopy to homology groups is defined and a homotopy operator, ∂, is determined by the condition h°∂=d°h. For general purposes, it might be an advantage to take the natural boundary operator as the standard one, in conformity with the practice in homology theory.