We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

This paper takes up the systematic study of the Gottlieb groups G n+k (S n ) of spheres for k ≤ 13 by means of the classical homotopy theory methods. We fully determine the groups G n+k (S n ) for k ≤ 13 except for the 2-primary components in the cases: k = 9, n = 53; k = 11, n = 115. In particular, we show [ι n , η 2The Gottlieb groups G k (X ) of a pointed space X have been defined by Gottlieb in [9] and [10]; first G 1 (X ) and then G k (X ) for all k ≥ 1. The higher Gottlieb groups G k (X ) are related in [10] to the existence of sectioning fibrations with fiber X . For instance, if G k (X ) is trivial then there is a crosssection for every fibration over the (k + 1)-sphere S k+1 , with fiber X . This paper grew out of our attempt to develop techniques in calculating G n+k (S n ) for k ≤ 13 and any n ≥ 1. The composition methods developed by Toda [36] are the main tools used in the paper. Our calculations also deeply depend on the results of [13,16,21].

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [ Σ ( V × W ∪ * ), X ] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan.

Let X be a 1-connected space with the homotopy type of a CW -space and H a finite group acting freely on X by homeomorphisms homotopic to the identity. We prove that l k η * G k (X) ⊆ G k (X/H) for all k > 1 and some estimated positive integer l k which depends on k, where G k is the k th Gottlieb group and η : X → X/H is the quotient map to the orbit space X/H. We show that l k is independent of k for X with the homotopy type of a finite CW -space. We also obtain that lπ k (X) ⊆ G k (X) for some positive integer l (independent on k) provided some restrictions are placed on the space X and the integer k > 1. Moreover, η

No abstract