It has been widely conjectured that there exists a homotopy commutative Diagram

where J is the stable Whitehead f-homomorphism and BSJ is the space constructed in [3]. In [4], Stasheff and the author proved that this conjecture is false. However, Quillen's proof of the Adams conjecture in [7] has as a corollary the existence of the homotopy commutative diagram

A number of unsolved problems of primal algebra theory concern the existence of certain collections of dependent primal algebras. In [3] E. S. O'Keefe showed that any collection of pairwise non-isomorphic primal algebras of type {n} with n > 1 forms a primal cluster. Recently the author has discovered that if τ is any type containing at least two elements, one of which is > 1, then there are at least two non-isomorphic dependent primal algebras of type τ, except possibly in the case = {2, 0}; this result will appear later.

The enumeration of the H-equivalence classes of multiplications on a space is a topic of current interest. In this paper we try to study the H-equivalence classes of multiplications on a CW complex X with finitely many non-vanishing homotopy groups, by using the Postnikov decomposition of X and multiplier arguments [1; 4], This paper presents a way to compute the set of H-equivalence classes of multiplications on X from the knowledge of certain quotient sets of H*(B Λ B, ∑) and some homotopy equivalences of B, where B represents the spaces in the Postnikov decomposition of X, and ∑ denotes abelian groups corresponding to the homotopy groups of X.

Computation of the velocity of a given motion depends on measurement of nearby position changes only. Computation of acceleration, on the other hand, depends on measurement of nearby changes in velocity. But since velocity vectors are attached to positions so that even nearby ones are not a priori comparable, acceleration is not computable until a rule for comparison of vectors along a curve is given. Such a rule-parallel translation or linear connexion - exists automatically in Euclidean spaces. For motions in more general manifolds, for example (semi-) Riemannian ones, parallel translation is a less obvious consequence of the metric properties.

Dans [4] C. S. Herz soulève la question suivante:

Soit G = SL(2, R) le groupe des matrices réelles d'order deux, de déterminant un; soit H = SL(2, Z) le sous-groupe discret de G, formé des matrices à coefficients entiers. Existe-t-il une constante absolue c > 0 telle que pour toute function f de At(H) à support fini, on puisse trouver une function de AP(G), à support compact dont la restriction à H coïncide a vec f et

A finite set of generators for the isotopy classes of selfhomeomorphisms of closed surfaces was given by Lickorish in three papers [2; 3; 4]. In [5] the group of isotopy classes for a particular, well-known cube with holes was presented. There the structure was “tight” enough to allow the computation of the relators as well as the generators. In this paper we give a finite set of generators for the group of isotopy classes of self-homeomorphisms on the solid double torus, the cube with two handles. Let us remark that the group of isotopy classes for the solid torus is well-known.

Let H denote the set of all entire functions of a single complex variable equipped with the topology of convergence uniform on all compact subsets of C, the set of complex numbers. Then an entire function f is mean periodic if the subspace spanned by f and its complex translates is not dense in H. It was shown by Schwartz [13, p. 922] in 1947, to whom this definition is due, that any such function is the limit in H of a certain sequence of exponential polynomials.

Soit L un groupe abélien élémentaire d'ordre pn (p est un premier, n est un entier positif). Soient (a) le groupe cyclique d'élément a ∈ L, \A\\e nombre des éléments de l'ensemble A ᑕ L. Nous appellerons l'ensemble B ᑕ L racine du groupe L si pour tout a, {0} 9 ≠ a ∈ L, il existe b1, b2 ∈ B (b1 ≠ b2) tels que b 1 b2 ∈ (a). (A ce propos, il est évident que si B est une racine de L, l'ensemble Bst* = {bi* = sbi + t/bi ∈ B}, où sont donnés s, t ∈ L, (s) ≠ (0), (c'est-à-dire Bst* est le résultat de l'affinité de B) est également racine de L.) Nous appellerons la racine B minimum et désignerons |B| par β (L) si tout autre racine contient au minimum |B| éléments. Voici les exemples de racines minimum du L pour L = F2n, n = 1, 2, 3, 4, 5 et pour L = F33:

The present investigation was stimulated by a theorem of Alfsen and Effros (4.9 of [1]) concerning a real Banach space, its ikf-ideals, and its primitive M-ideals (these are denned in [1]). This theorem states that a real Banach space is in the natural way a module over the ring of bounded continuous real-valued functions on the space of primitive Jlf-ideals with the Jacobson topology.

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

In [1], C. C. Chen and G. Grâtzer developed a method for studying Stone algebras by associating with each Stone algebra L, a uniquely determined triple (C(L), D(L), ɸ (L)), consisting of a Boolean algebra C(L), a distributive lattice D(L), and a connecting map ɸ(L). This approach has been successfully exploited by various investigators to determine properties of Stone algebras (e.g. H. Lakser [9] characterized the injective hulls of Stone algebras by means of this technique). The present paper is a continuation of this program.

In this paper we construct two examples which elucidate the relationships between several σ-algebras that arise in measure-theoretic constructions on locally compact spaces and groups. For any space X let (X) be the Borelσ-algebra on X, i.e., the smallest σ-algebra of subsets of X which contains the family of all closed subsets of X. Let δ (X) be the smallest δ-ring of subsets of X which contains every compact subset of X, where by a δ-ring we mean a collection of subsets of X which is closed under the formation of countable intersections, finite unions and relative complements. Let σ(X) be the smallest σ-ring of subsets of X which contains all compact subsets of X, where by a σ-ring we mean a collection of subsets of X which is closed under the formation of countable unions, finite intersections and relative complements.

Quasi-injective and quasi-projective modules over hereditary noetherian prime rings ((hnp)-rings) were studied in [17]. In the present paper we give some applications of the results established in [17]. Kulikov, Kertesz, Prufer, Szele had made basic contributions to the problem of decomposability of abelian p-groups (Fuchs [4]). Kaplansky [9] studied analogous problems for modules over (commutative) Dedekind domains. Let R be an (hnp)-r'mg, which is not right primitive. Using the structure of an indecomposable infective torsion R-module, established in [17, Theorem 4], some of the basic concepts and results on the decomposability of a torsion abelian group are generalized in Section 2, to modules over R.

This article continues the study of Abian's order on commutative semiprime rings (for such a ring R, the relation a ≦ b if and only if ab = a2” makes R into a partially ordered multiplicative semigroup). The aim, here, is to extend as far as possible the theorem of Brainerd and Lambek which says that the completion of a Boolean ring is its complete ring of quotients. Only certain subsets of a ring may have upper bounds (in any extension ring) and these are called boundable (the notion is due to Haines). A ring will be called complete if every boundable subset has a supremum. If R ⊂ S are (commutative semiprime) rings then S will be called a completion of R if S is complete and every element of S is the supremum of a subset of R.

A non-Hausdorff Ascoli theorem for continuous functions was established in [6]. The present purpose is to extend this result to point-compact continuous multifunction, using Levine's generalization for closed subsets [12]. The paper is organized as follows: the object of section 2 is to establish the necessary multifunction lemmas and to introduce the notion of a Tychonoff set; section 3 generalizes to multifunction context the partial exponential law of R. H. Fox [9, p. 430], and establishes a special exponential law for multifunctions; section 4 concerns the crucial properties of even continuity for multifunctions, introduced in [8]; the main theorem of the paper is established in section 5.

Classically CW-complexes were found to be the best suited objects for studying problems in homotopy theory. Certain numerical invariants associated to a CW-complex X such as the Lusternik-Schnirelmann Category of X, the index of nilpotency of ᘯ(X), the cocategory of X, the index of conilpotency of ∑ (X) have been studied by Eckmann, Hilton, Berstein and Ganea, etc. Recently D. G. Quillen [6] has developed homotopy theory for categories satisfying certain axioms. In the axiomatic set up of Quillen the duality observed in classical homotopy theory becomes a self-evident phenomenon, the axioms being so formulated.

This paper deals with some applications in the results obtained in “Numerical Invariants in Homotopical Algebra” [7]. The applications are mainly concerned with the homotopy theory of modules developed by P. J. Hilton [4]. However we have to restrict the class of rings because we want to obtain a situation where the axioms of Quillen [6] hold good. This paper is organised as follows.