Homotopy Groups and Torus Homotopy Groups

In recent years it has been found that the homotopy groups [9] can be made into a more powerful tool by the introduction of new operations. Two such have been defined and utilized: (i) every element of the fundamental group 7r, induces an automorphism of the n-dimensional homotopy group 7rX, so that 7rn becomes a group with operators; (ii) to every pair of elements a of Trm and ,3 of 7n there is defined an element a o fl the Whitehead product of a and i3, which belongs to the group 7r.m+?. It is shown in this paper that the homotopy groups of various dimensions and the operations (i) and (ii) can all be welded together into a single group r which I call the torus homotopy group. Let us sketch briefly the process of unification. In the first place the algorithms (4.2) given by J. H. C. Whitehead [12] for calculating with the operation o are strongly reminiscent of rules for manipulating commutators. Moroever, in the only case in which the Whitehead product can be compared with the group operation-the case m = n = m + n -1 = 1-the Whitehead product a o $ is identical with the commutator [a, 1 = aca1 /-1. Furthermore M. Abe [1] has defined, for every integrer n > 2, a group Kn , containing subgroups isomorphic to wi and to 7rn, in which the operation of 7r on 7rn becomes an inner automorphism. Finally (cf ?4) the operation (i) may be described in terms of (ii) and Abe's description of (i) as an inner automorphism implies that a o ,3 is a commutator whenever either m or n is equal to 1. These facts suggested to me that the Whitehead product might be, without restriction on either m or n, an actual commutator operation in a suitably defined group. Thus I was led to define, for every positive integer r, a group Tr which I call the r-dimensional torus homotopy group [7]. This group contains subgroups isomorphic to 7n for every n 1.) The torus homotopy groups are defined in a natural way, closely resembling the definitions of the homotopy and Abe groups. As might be expected the Abe groups Kn, 2 < n < r, are also contained isomorphically in Tr (?10). The finite dimensional torus homotopy groups r, are all contained isomorphically in one huge group r, the torus homotopy group. Perhaps the most remarkable consequence of the definition of these new groups is a disproof of the widely held belief that "all of the non-Abelian character of a space is expressed in its fundamental group." In fact there are an infinite number of simply connected spaces which have non-Abelian torus homotopy groups. This state of affairs is an immediate consequence of the non-triviality of certain Whitehead products in a certain type of simply-connected space [12].