No abstract

Introduction. In this paper we investigate an operation which is a generalization of the Whitehead product for homotopy groups. Let π(R, S) denote the collection of homotopy classes of base point preserving maps of R into S, let ΣR denote the reduced suspension of R, and let R ^ S be the identification space R x S/R V S (see § 1 for complete definitions). Then this generalized Whitehead product (written GWP) assigns to each a e π(ΣA, X) and β e π(ΣB, X) an element [a, β] e π(Σ (A & B), X), where A and B are polyhedra and X is a topological space. In the case when A and B are spheres [a, β] is essentially the Whitehead product. In this paper we generalize known results on spheres and Whitehead products to polyhedra and GWPs.The paper is divided into six parts. After the preliminaries of § 1 we present two definitions of the GWP in § 2. The first definition, which was given by Hilton in [8 pp. 130-131], is closely related to a commutator of group elements. The second definition is essentially a generalization of the ordinary Whitehead product. It first appeared, stated in the language of carrier theory, in a paper by D. E. Cohen [3]. We prove in Theorem 2.4 that these two definitions coincide. This generalizes a result of Samelson [11 p. 750].In § 3 we establish some properties of the GWP such as anti-commutativity and bi-additivity. With the exception of Proposition 3.1 the results of this section have been obtained by Cohen [3]. However, the proofs that we give are based on the first definition and facts about commutators. Moreover, we believe that our proofs are quite elementary.In the next section we show that ΣA x ΣB has the same homotopy type as the space obtained by attaching a cone by means of the GWP map. We then deduce a few simple consequences of this. In § 5 we consider the different ways that the GWP may be trivial. We study the following situations: (i) [α, β] -0 (ii) the GWP map is nullhomotopic (iii) X is a space in which all GWPs vanish. With regard to (iii) we see that such spaces are not necessarily iJ-spaces.The final section is devoted to a product which is dual (in the sense of Eckmann and Hilton) to the GWP. Two definitions of the dual product are given and they are shown to be equivalent. We also indicate some properties of the dual product.

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V 0 ⊕ V 1 and that f and g are homotopic on ΛV 0 . An obstruction is then obtained as a vector space homomorphism V 1 → H * (N ). We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes [M, N ]. This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A, B] when A and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps.Mathematics Subject Classification (1991):55P62, 55Q05, 55S35, 55P10

No abstract

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.

The theory of minimal models, as developed by Sullivan [6; 8; 16] gives a method of computing the rational homotopy groups of a space X (that is, the homotopy groups of X tensored with the additive group of rationals Q). One associates to X a free, differential, graded-commutative lgebra , over Q, called the minimal model of X, from which one can read off the rational homotopy groups of X.

No abstract

For a based, 1-connected, finite CW-complex X, we study the following subgroups of the group of homotopy classes of self homotopy equivalences of X: E * (X), the subgroup of homotopy classes which induce the identity on homology groups, E * (X), the subgroup of homotopy classes which induce the identity on cohomology groups and E dim+r # (X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ≤ dim X + r. We investigate these groups when X is a Moore space and when X is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on X such that E * (X) = E * (X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine E dim+r # (X) for certain spaces X and completely determine the group when X = S m × S n and X = CP n ∨ S 2n. 1991 Mathematics Subject Classification. Primary 55P10. Key words and phrases. Homotopy equivalences, the group of homotopy equivalences, homotopy equivalences which induce the identity, Moore spaces, co-Moore spaces, products of spheres.