Topological Group Research Articles

Generic properties of topological groups

Abstract We study generic properties of topological groups in the sense of Baire category. First, we investigate countably infinite groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. Recently, I. Goldbring, S. Kunnawalkam Elayavalli, and Y. Lodha proved that every isomorphism class is meager among countably infinite groups. In contrast, it follows from the work of W. Hodges on model-theoretic forcing that there exists a comeager isomorphism class among countably infinite abelian groups. We present a new elementary proof of this result. Then, we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.

Mitochondrial Genome and Phylogenetic Analysis of the Narrownose Smooth-Hound Shark Mustelus schmitti Springer, 1939

Southern Brazil is home to a large biodiversity of elasmobranchs from the Brazilian coast. Several genera and species of small sharks of the Triakidae family live in this marine environment. Studies on these shark species are scarce, with few genetic data and little information on animal population structures. The present study aimed to sequence the complete mitochondrial genome (mtDNA) of the endangered species Mustelus schmitti (narrownose smooth-hound shark) and to perform a phylogenetic analysis of the Triakidae family. The mtDNA sequenced here was 16,764 bp long and possessed the usual 13 mitochondrial protein coding genes (PCGs), 22 tRNAs, two rRNAs (12S and 16S) and a large D-loop DNA sequence, presenting an overall organization similar to other species from the genus Mustelus. Phylogenetic analyses were performed using a dataset containing this new mtDNA and 59 other mitochondrial genomes of the Carcharhiniformes species (including 14 from the Triakidae family), using the Maximum Likelihood (ML) method. All the species of the Triakidae family were clustered into a monophyletic topology group. In addition, polyphyly was observed in Galeorhinus galeus, Hemiatrakis japanica, Triakis megalopterus and Triakis semifasciata. In conclusion, this study contributes to a deeper understanding of the genetic diversity of sharks and represents an important step towards the conservation of these endangered animals.

Numerical Methods for Topological Optimization of Wooden Structural Elements

Timber represents a building material that aligns with the environmental demands on the impact of the construction sector on climate change. The most common engineering solution for modern timber buildings with large spans is glued laminate timber (glulam). This project proposes a tool for a topological optimized geometry generator of structural elements made of glulam that can be used for building a database of topologically optimized glulam beams. In turn, this can be further used to train machine learning models that can embed the topologically optimized geometry and structural behavior information. Topological optimization tasks usually require a large number of iterations in order to reach the design goals. Therefore, embedding this information into machine learning models for structural elements belonging to the same topological groups will result in a faster design process since certain aspects regarding structural behavior such as strength and stiffness can be quickly estimated using Artificial Intelligence techniques. Topologically optimized geometry propositions could be obtained by employing generative machine learning model techniques which can propose geometries that are closer to the topologically optimized results using FEM and as such present a starting point for the design analysis in a reduced amount of time.

On Ψω-factorizable groups

A topological group G is called Ψω-factorizable (resp. M-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with ψ(H)≤ω (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of Ψω-factorizable groups. It is shown that a topological group G is Ψω-factorizable if and only if every continuous real-valued function on G is Gδ-uniformly continuous, if and only if for every cozero-set U of G, there exists a Gδ-subgroup N of G such that UN=U. Sufficient conditions on the Ψω-factorizable group G to be M-factorizable are that G is τ-fine and τ-steady for a cardinal τ.

Local permutation stability

We introduce a notion of “local stability in permutations” for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups. Our notion is weaker than the “permutation stability” introduced by Glebsky–Rivera and Arzhantseva–Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for a finitely generated amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky, and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng’s classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.

Discrete reflexivity in topological groups and function spaces

Discrete reflexivity in topological groups and function spaces

Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups

We study two topological properties weaker than feeble compactness in the class of (para)topological groups, the compact-boundedness and weak compact-boundedness, both introduced by Angoa, Ortiz-Castillo and Tamariz-Mascarúa in [2]. First, given a subgroup H of a topological group G, we show how to extend these properties from the quotient space G/H to G; this, in the cases when H is a compact, locally compact or (weakly) compact-bounded subgroup. Secondly, we prove the main result of this article: if a Tychonoff space X is compact-bounded and not scattered, then the free topological group F(X) and the free Abelian topological group A(X) admit a non-trivial metrizable quotient group; thus extending Theorem 4.7 by Leiderman and Tkachenko in [15]. Finally, we study the r-weakly compact-bounded subsets of a topological space X. We show that r-weak compact-boundedness is a productive property. Moreover, sufficient conditions are given in order for a C-compact subset of a paratopological group G to become an r-weakly compact-bounded subset. This article is part of a larger work developed in [16] and [17].

Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups

Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups

THE SHAPE OF COMPACT COVERS

Abstract For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ , and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa. We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces. Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $ -pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$ .

Diffeomorphisms of Foliated Manifolds I

The set $Diff(M)$ of all diffeomorphisms of manifold $M$ onto itself is the group related to composition and inverse mapping. The group of diffeomorphisms of smooth manifolds is of great importance in differential geometry and analysis. It is known that the group $Diff(M)$ is topological group in compact open topology.In this paper we investigate the group $Diff_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. In this paper we prove that if all leaves of the the foliation $F$ are closed subsets of $M$ then the foliated compact open topology of the group $Diff_{F}(M)$ coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold is studied when foliation generated by riemannian submersion.

NMR residual dipolar couplings investigation in the topology of house dust mite Group V allergens

Blo t 5 is an important major allergen protein from Blomia tropicalis mites, which are prevalent in tropical and subtropical regions, including Taiwan. It is a coiled-coil triple helical bundle, but there currently is ambiguity around its structural fold and packing of the three helices. We have relied on NMR residual dipolar coupling data collected from four different alignment media to confirm that Blo t 5 has left-handed helical topology and further used that data to refine its solution structure. Earlier we had described conformational epitope for a detection monoclonal antibody by exclusive use of TROSY NMR experiments that studied Blo t 5 binding with the antibody FAB’ fragment. Here, we confirm those findings with an extensive mutagenesis and biophysical study to validate the NMR epitope mapping approach proposed by us.

Galois theory and homology in quasi-abelian functor categories

Given a finite category T, we consider the functor category AT, where A can be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories AT for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of AT whose objects are given by the functors F:T→A with F(T)=0 for all T∉S. We prove that F is a torsion-free Birkhoff subcategory of AT. This allows us to study (higher) central extensions from categorical Galois theory in AT with respect to F and generalized Hopf formulae for homology.

All Hartman–Mycielski groups are minimally almost periodic

For each topological group G, it is shown that the Hartman–Mycielski group G• over G is minimally almost periodic, that is, all continuous homomorphisms of G• to compact Hausdorff topological groups are trivial. We also show that the group G•, over a compactly generated topological group G, is algebraically generated by a compact connected subset. Consequently, a wealth of minimally almost periodic, connected, compactly generated topological Abelian groups exist.

Some characterizations of ω-balanced topological groups with a q-point

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.

On generalized Milnor's conjecture for abelian topological groups

We consider an extension of Milnor's conjecture [14] from Lie groups to arbitrary topological groups and prove its generalized version for locally compact finite dimensional abelian groups.We present counterexamples to extended Milnor's conjecture in the class of abelian topological groups whose underlying space is a CW-complex.

Group topologies on the integers generated by sequences

In the first part of the paper we study various ways to introduce a group topology on the group Z of integers by using sequences of natural numbers b, among them the well-known Protasov-Zelenyuk topology Tb for a T-sequence b. We prove, among others, that for slowly diverging sequences b, the group (Z,Tb) is not locally quasi-convex, in particular not reflexive. As an application, in the second part we resolve Problem 6.1 and Problem 6.5 from [28].

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Rigid topologies on groups

Our main result is to show that every infinite, countable, residually finite group G admits a Hausdorff group topology which is neither discrete nor precompact.

On Soft Topological Groups and Soft Function Spaces

In this paper, the definition of a soft topological group based on the concept of soft points is introduced. Let [Formula: see text] denote the collection of all soft continuous functions between soft topological spaces [Formula: see text] and [Formula: see text]. We investigate the structures of abstract groups, and soft groups within the context of the soft function space. Furthermore, the soft topological group structure of the group [Formula: see text] under soft topologies is explored.

Isotropy group on some topological transformation group structures

This paper explores the topological properties of irresolute topological groups, their quotient maps, and the role of topology in normal subgroups. It provides a detailed analysis\linebreak using examples and counterexamples. The study focuses on the essential features of irresolute topological groups and their quotient groups, for understanding the topological aspects of isotropy groups. For a trans\-for\-ma\-tion group $(\mathsf{H}, \mathsf{Y}, \psi)$ and a point $y \in \mathsf{Y},$ the set \centerline{$\mathsf{H}_{y} = \{h \in \mathsf{H} \colon hy = y\}$} \noi consisting of elements of $\mathsf{H}$ that fix $y$, is called the isotropy group at $y$. The paper highlights the distinct topological characteristics of isotropy groups in transformation group structure. It demonstrates that if $(\mathsf{H}, \mathsf{Y}, \psi)$ is an Irr$^{*}$-topological transformation group, then $( \mathsf{H}/ \mathop{Ker} \psi, \mathsf{Y}, \overline{\psi})$ forms an effective Irr$^{*}$-topological transformation group. By investigating both irresolute topological groups and isotropy groups, the study provides a clear understanding of their topological features. This research improves our understanding of these groups by offering clear examples and counterexamples, leading to a thorough conclusion about their different topological features.