On the lifting of structure groups

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

The fineness index of topological groups and [formula omitted]-factorizable groups

Hattori topologies on almost topological groups

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spectrum of the crossed product by the stabilizer group bundle. The main theorem is also generalized in the groupoid algebra case to an identification of the primitive ideal space. This generalization replaces the assumption that the orbit space is well behaved with an amenability hypothesis. We then use induction to show that the primitive ideal space of the groupoid algebra is homeomorphic to a quotient of the dual of the stabilizer group bundle. In both cases the identification is topological. We then apply these theorems in a number of examples, and examine when a groupoid algebra has Hausdorff spectrum. As a separate result, we also develop a theory of principal groupoid group bundles and locally unitary groupoid actions. We prove that such actions are characterized, up to exterior equivalence, by a cohomology class which arises from a principal bundle. Furthermore, we also demonstrate how to construct a locally unitary action from a given principal bundle. This last result uses a duality theorem for abelian group bundles which is also included as part of this thesis.

All Hartman–Mycielski groups are minimally almost periodic

A note on multiplier convergent series

Let A be a lattice-ordered group. Gusic showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusic’s theorem, and reveal the very nature of a “C-group” of Gusic in this paper. Moreover, we show that the C-topological groups are topological lattice-ordered groups, and prove that every archimedean lattice-ordered vector space is a T2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T2. A further example demonstrates that a T2 topological archimedean lattice-ordered group need not be C-archimedean, either.

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

Among the twenty-three mathematical problems that Hilbert posed at the Second International Congress of Mathematicians in Paris in 1900 (see [10]), the fifth problem is concerned with Lie’s theory of transformation groups, and in its second part deals with what Hilbert calls “infinite groups,” but which are not groups in the modern use of the term. The questions in this second part of the fifth problem concern functional equations and difference equations and have, for example, connections with the work of N.H. Abel. These questions lie completely outside the theory of transformation groups, and we do not discuss them here any further. We refer the reader to [1] for the “state of the art” of 1989 in this area of research. We recall that a topological transformation group consists of a topological group G, a topological space X, and a continuous action of G on X, that is, a continuous map

Long colimits of topological groups II: Free groups and vector spaces

Incomplete infima of complete uniformities related to topological groups and topological vector spaces

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.