On Free p‐Compact Groups

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.

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.

All Hartman–Mycielski groups are minimally almost periodic

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

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

On the discontinuity of the π1-action

Let (X,+) be a topological abelian group. We discuss regularity of solutions f:X→R of Hlawka’s functional inequality f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z), postulated for all x,y,z∈X. We study the lower and upper hull of f. Moreover, we provide conditions which imply continuity of f. We prove, in particular, that if X is generated by any neighborhood of zero, f is continuous at zero, and f(0)=0, then f is continuous on X.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

Duality theory for convergence groups

Weakly complete free topological groups

The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg (G): If G is a locally compact Abelian group with Bohr compact- ification bG, and if N is a closed metrizable subgroup of bG, then every A G satisfies: A · (N G) is compact in G if and only if {aN : a 2 A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T , with U totally bounded, T locally compact, U T , with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of hG,T i. There are applications to topological groups of the form kG for G a totally bounded Abelian group.

Topological groups with thin generating sets

Compactness and connectedness in topological groups

Characterization of almost maximally almost-periodic groups

Let G be a compact connected group and let H be a locally compact abelian group. Denote by C e ( G , H ) {C_e}(G,H) the space of all identity preserving continuous functions from G to H with the compact-open topology, and denote by Hom(G, H) the space of all homomorphisms in C e ( G , H ) {C_e}(G,H) . We prove that C e ( G , H ) {C_e}(G,H) is isomorphic to V × Hom ( G , H ) V \times {\text {Hom}}(G,H) , where V is a topological vector space. This is used to prove that every element of C e ( G , H ) {C_e}(G,H) is homotopic to precisely one element of Hom(G, H). We also prove that the fundamental group of H is isomorphic to Hom(K, H), K being the circle group, that π n ( H ) = 0 {\pi _n}(H) = 0 for n ≧ 2 n \geqq 2 , and that a compact connected abelian group admits essentially only one H-space structure.