On Products in Homotopy Groups

Abstract

One of the outstanding problems in homotopy theory is that of determining the homotopy groups of simple spaces. Even for as simple a space as the n-sphere very little is known. In fact, in most cases, it is not known whether or not the homotopy groups are zero. J. H. C. Whitehead [16]2 has defined a product between two of the homotopy groups 7r, and 7rq of a space X with values in 7rp+q-. In some instances this product affords a method for constructing non-zero elements of 7rp+,q (X). He also defined generalized products, involving the homotopy groups of the rotation groups. Hurewicz [10] originally defined the group 7rn(X) as the fundamental group of a certain function space over X. The elements of 7rn(X) may also be regarded as equivalence classes of mappings of the n-sphere Sn into X, and it is the latter definition which has been used in most of the applications. In this paper the original point of view adopted by Hurewicz is combined with the second approach. The method of fibre spaces of Hurewicz and Steenrod [11] is used to study the interrelations between the homotopy groups of a space X and those of certain function spaces over X. In Section 2 we state preliminary results, many of which are known, and establish the necessary homomorphisms between the homotopy groups of the spaces under consideration. In Section 3 the products of J. H. C. Whitehead are characterized as operations in function spaces. Using this characterization, we are able to prove that the Freudenthal Einhingung of a product is always inessential. A partial converse to this result is obtained. In Section 5 the generalized products defined by J. H. C. Whitehead are characterized in terms of known operations. This characterization is used to verify a conjecture of J. H. C. Whitehead.