Topological Group Structure Research Articles

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.

Topologies on the real line

We prove that if a topology on the real line endows it with a topological group structure (additive) for which the interval (0,+infty ) is an open set, so this topology is stronger than the usual topology. As a consequence we obtain characterizations of the usual topology as group topology and as ring topology. We also proved that if a topology on the real line is compatible with its usual lattice structure and is T_1, so this topology is stronger than the usual topology, and as a consequence we obtain a characterization of the usual topology as lattice topology.

General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes

Positive self-similar Markov processes are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in what happens if we consider Markov processes in dimension 1 or 2 that satisfy self-similarity properties of a more general form than a scaling property. We characterize them by proving a generalized Lamperti representation. Our results show that, in dimension 1, the classical Lamperti representation only needs to be slightly generalized. However, in dimension 2, our generalized Lamperti representation is much more different and involves the exponential functional of a bivariate Lévy process. We briefly discuss the complications that occur in higher dimensions. We present examples in dimensions 1, 2 and 3 that are built from growth-fragmentation, self-similar fragmentation and Continuous-state Branching processes in Random Environment. Some of our arguments apply in the context of a general state space and show that we can exhibit a topological group structure on the state space of a Markov process that satisfies general self-similarity properties, which allows to write a Lamperti-type representation for this process in terms of a Lévy process on the group.

A group-theoretic characterization of the Fock-Bargmann-Hartogs domains

Let $M$ be a connected Stein manifold of dimension $N$ and let $D$ be a Fock-Bargmann-Hartogs domain in $\mathbb{C}^N$. Let $\mathrm{Aut}(M)$ and $\mathrm{Aut}(D)$ denote the groups of all biholomorphic automorphisms of $M$ and $D$, respectively, equipped with the compact-open topology. Note that $\mathrm{Aut}(M)$ cannot have the structure of a Lie group, in general; while it is known that $\mathrm{Aut}(D)$ has the structure of a connected Lie group. In this paper, we show that if the identity component of $\mathrm{Aut}(M)$ is isomorphic to $\mathrm{Aut}(D)$ as topological groups, then $M$ is biholomorphically equivalent to $D$. As a consequence of this, we obtain a fundamental result on the topological group structure of $\mathrm{Aut}(D)$.

Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representation

Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representation

Topological group structure of neutrosophic set

The notion of neutrosophic normal subgroup are defined and some of their properties are discussed. Further we initiate the concept of neutrosophic topological groups and related results have been derived.

Extended topological group structure due to average reflection symmetry

We extend the single-particle topological classification of insulators and superconductors to include systems in which disorder preserves average reflection symmetry. We show that the topological group structure of bulk Hamiltonians and topological defects is exponentially extended when this additional condition is met and examine some of its physical consequences. Those include localization–delocalization transitions between topological phases with the same boundary conductance as well as gapless topological defects stabilized by average reflection symmetry.

ONE DIMENSIONAL T.T.T STRUCTURES

AbstractIn this paper we analyze the relationship between o-minimal structures and the notion of ω-saturated one-dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one-dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn’t split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an ω-saturated one-dimensional t.t.t structure admits a topological group structure then it is locally o-minimal. We also prove that the number of connected components in a definable family is uniformly bounded, which implies that an elementary extension of an ω-saturated one-dimensional t.t.t structure is t.t.t as well.

More on generalized topological groups

In the paper [Hussain, M., Khan, M. and Ozel, C., ¨ On Generalized Topological Groups] we defined the generalized topological group structure and we proved some basic results. In this work we introduce the notions of ultra Hausdorffness and ultra G-Hausdorffness and we give the relation between the ultra G-Hausdorffness and G-compactness.

Covering maps over solenoids which are not covering homomorphisms

Let Y be a connected group and let f:X->Y be a covering map with the total space X being connected. The following question is considered: Is it possible to define a topological group structure on X making f a homomorphism of topological groups. The question is answered in the negative by constructing a connected infinite-sheeted covering map over an arbitrary solenoid which is not an overlay map and consequently not a covering homomorphism.

The fundamental group as a topological group

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted π1τ, takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology.

Homotopy normal maps

Normal maps between discrete groups $N\rightarrow G$ were characterized [FS] as those which induce a compatible topological group structure on the homotopy quotient $EN\times_N G$. Here we deal with topological group (or loop) maps $N\rightarrow G$ being normal in the same sense as above and hence forming a homotopical analogue to the inclusion of a topological normal subgroup in a reasonable way. We characterize these maps by a compatible simplicial loop space structure on $Bar_\bullet(N,G)$, invariant under homotopy monoidal functors, e.g. Localizations and Completions. In the course of characterizing homotopy normality, we define a notion of a "homotopy action" similar to an $A_{\infty}$ action on a space, but phrased in terms of Segal's 'special $\Delta-$spaces' and seem to be of importance on its own right. As an application of the invariance of normal maps, we give a very short proof to a theorem of Dwyer and Farjoun namely that a localization by a suspended map of a principal fibration of connected spaces is again principal.

Continuous interval exchange actions

Let E denote the group of all interval exchange transformations on [0,1). Given a suitable topological group structure on E, it is possible to classify all one-parameter interval exchange actions (continuous homomorphisms from R to E). In particular, up to conjugacy in E, any one-parameter interval exchange action factors through a rotational torus action.

Characterizing group C*-algebras through their unitary groups: the Abelian case

We study to what extent group C∗-algebras are characterized by their unitary groups. A complete characterization of which Abelian group C∗-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of C∗(Γ), such as the K-theoretic K1(C∗(Γ)) and find that C∗-algebras of nonisomorphic torsion-free groups may have isomorphic K1-groups, in sharp contrast with the well-known fact that C∗(Γ) (even Γ) is characterized by the topological group structure of its unitary group when Γ is torsion-free.

Algebraic classification of physical structures with zero. II. Topological aspects

Some topological analogs are obtained of the author’s previous results on the classification of physical structures. A topological-algebraic axiomatics is considered enabling us to replace the algebraic axiom corresponding to the main equation of a physical structure by a more natural axiom. A physical structure of rank different from (2, 2) is shown to be a pair of vector spaces with a nondegenerate bilinear form over a topological skew field. The obtained results are applied to the classification of physical structures of rank different from (2, 2). The structures of rank (2, 2) are also considered. To describe them, a topological group structure corresponding to the biform action is introduced on the set of measurements.

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities alg-rk(W) ≤ flat-rk(G) ≤ rk(|W|0). Here, alg-rk(W) is the maximal Z-rank of abelian subgroups of W, and rk(|W|0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|0. We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.

On the structure of finite coverings of compact connected groups

Finite-sheeted covering mappings onto compact connected groups are studied. We show that for a covering mapping from a connected Hausdorff topological space onto a compact (in general, non-abelian) group there exists a topological group structure on the covering space such that the mapping becomes a homomorphism of groups. To prove this fact we construct an inverse system of covering mappings onto Lie groups which approximates the given covering mapping. As an application, it is shown that a covering mapping onto a compact connected abelian group G must be a homeomorphism provided that the character group of G admits division by degree of the mapping. We also get a criterion for triviality of coverings in terms of means and prove that each finite covering of G is equivalent to a polynomial covering.

Smooth Lie groups over local fields of positive characteristic need not be analytic

We describe finite-dimensional smooth Lie groups over local fields of positive characteristic which do not admit an analytic Lie group structure compatible with the given topological group structure, and C n -Lie groups without a compatible C n + 1 -Lie group structure for each n ⩾ 1 . We also present examples of non-analytic, smooth automorphisms of analytic Lie groups over such fields, as well as C n -automorphisms which fail to be C n + 1 .

Diagonal structures on topological spaces

We investigate when a topological space admits a partial product operation satisfying some rather weak continuity restrictions and almost nothing else-the only algebraic requirement is that some element e of X is a left and a right identity with respect to this multiplication. The operation is called partial diagonalization of X at e. Several sufficient conditions for a space to be partially diagonalizable are established. On the other hand, it is shown that certain deep results about the topological structure of compact topological groups can be extended to partially diagonalizable compact spaces. We also discover that partial diagonalizability plays an important role in the theory of cardinal invariants, in the study of homogeneous spaces, and in such classical topics of general topology as the theory of Stone–Čech compactification and the theory of Hewitt–Nachbin compactification. The notions of a Moscow space and of a C-embedding are instrumental in our study.

A generalization of Hagopian's theorem and exponents

We generalize Hagopian's theorem characterizing solenoids to higher dimensions by showing that any homogeneous continuum admitting a fiber bundle projection onto a torus with totally disconnected fibers admits a compatible topological group structure. And then the higher dimensional exponent group is introduced.