Binary operations for homotopy groups with coefficients

We define and study binary operations for homotopy groups with coefficients, and give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of the generalized Whitehead product to these operations. We discuss two classes of binary operations, i.e., the Whitehead products and the torsion products. We also introduce a new class of operations called Ext operations and determine some of its properties. Then we compare the torsion product to the Whitehead product in a special case, and prove that the smash product of two Moore spaces has the homotopy type of a wedge of two Moore spaces.     

Categories equivalent to the category of rational H-spaces

The category of rationalH-spaces is shown to be equivalent to the category of commutative Hopf algebras over , the category of cocommutative Hopf algebras over , and the categoryL of graded Lie algebras over by the rational cohomology, homology, and homotopy functors, respectively. Several consequences of these equivalences are derived. It is also proved that the loop-space functor is an equivalence from the category of coformal rational spaces to . Dually, the category of rationalcoH-spaces is shown to be equivalent to the comonoid category ofL and to the category of cocommutative coalgebras over . The suspension functor is an equivalence from the category of formal, rational spaces to the category of 2-connected, rationalcoH-spaces.     

Equivalent definitions of the Ganea fibrations¶and cofibrations

The Ganea fibrations have been used to define Lusternik–Schnirelmann category. It is proved that three definitions of the Ganea fibrations are equivalent. Consequences are drawn for the category of a map and the notion of cocategory. In addition, Ganea cofibrations are briefly considered. Received: 2 April 1999     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . 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 . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA 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. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

The group of homotopy equivalences of products of spheres and of Lie groups

We investigate the group of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of provided the p-localization of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that is a semidirect product of certain homotopy groups . We also show that has a central series whose successive quotients are , which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of and give numerous explicit calculations. Received: 14 April 2001; in final form: 10 September 2001 / Published online: 17 June 2002     

Comultiplications in algebra and topology

Si studiano le proprietà degli oggetti con comoltiplicazione (cioè co-H-oggetti) nelle categorie che ammettono un coprodotto e i funtori fra queste categorie. Si dimostra che questi funtori conservano le proprietà dei co-H-oggetti. Queste osservazioni generali si applicano inoltre ai seguenti esempi specifici di categorie e funtori: la categoria omotopica degli spazi puntati, la categoria delle algebre graduate commutative, la categoria delle algebre graduate associative, la categoria delle algebre di Lie graduate e la categoria dei gruppi. I funtori sono: il gruppo fondamentale, il funtore coomologia, il funtore omologia dello spazio di cammini chiusi di uno spazio e il funtore gruppo omoto pico. Si dimostra che in questi casi specifici si ottengono risultati classici ben noti.

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Conferenza tenuta il 15 aprile 1996     

Properties of comultiplications on a wedge of spheres

In this paper we study the set of comultiplications on a wedge of a finite number of spheres. We are interested in group theoretic properties of these comultiplications such as associativity and commutativity and loop theoretic properties such as inversivity, power-associativity and the Moufang property. Our methods involve Whitehead products in wedges of spheres and the Hopf-Hilton invariants. We obtain extensive results for a restricted class of comultiplications, namely, the one-stage quadratic or cubic comultiplications.     

The inverses of an H-space

 A multiplication on an H-space X has a left inverse λ and a right inverse ρ. They are mutual inverses and λ = ρ if and only if . In this paper we investigate the order of λ. We give an example of a multiplication with , and prove that for any finite H-complex X there are finitely many left inverses of finite order. Conditions are given for there to be infinitely many multiplications on X with the same left inverse. We then give conditions for a left inverse to have infinite order. We apply these results to specific Lie groups. Received: 15 June 2001