scholarly journals Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1421
Author(s):  
Susmit Bagchi
Keyword(s):  
Cyclic Group ◽  
Group Structure ◽  
Covering Space ◽  
Group Homomorphism ◽  
Connected Components ◽  
Homotopy Equivalence ◽  
Topological Groups ◽  
Covering Spaces ◽  
Multiple Identity ◽  
Path Homotopy

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

scholarly journals More on generalized topological groups

2013 ◽  
Vol 22 (1) ◽  
pp. 47-51
Author(s):  
MURAD HUSSAIN ◽  
◽  
MOIZ UD DIN KHAN ◽  
CENAP OZEL ◽  
◽  
...  
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
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.

VAGUE GROUPS AND Ω-VAGUE GROUPS

2005 ◽  
Vol 01 (02) ◽  
pp. 229-242 ◽  
Author(s):  
KIRAN R. BHUTANI ◽  
JOHN N. MORDESON
Keyword(s):  
Direct Product ◽  
Group Structure ◽  
Group Homomorphism ◽  
Fuzzy Subset ◽  
Product Of Subgroups ◽  
A Chain

Given a group G, we show how one can define a vague group structure on G via a chain of subgroups of G. We discuss how a group homomorphism f from a vague group X onto a group Y induces a vague group structure on Y with f satisfying the vague homomorphism property. The notion of Ω-vague groups is introduced, where Ω is a fuzzy subset. The direct product G1 × G2 of two vague groups and the internal vague direct product of subgroups of a vague group is introduced.

scholarly journals Limits of covering spaces and residual properties of groups

1994 ◽  
Vol 36 (3) ◽  
pp. 277-282
Author(s):  
Jon Michael Corson
Keyword(s):  
Covering Space ◽  
Covering Spaces ◽  
Residually Finite ◽  
Residual Properties ◽  
Space Approach

The purpose of this paper is to point out a flaw in H. B. Griffiths' covering space approach to residual properties of groups [3]. One is led to this paper from Lyndon and Schupp's book [4, pp. 114, 141] where it is cited for covering space methods and a proof that F-groups are residually finite. However the main result of [3] is false.

scholarly journals A presentation for a group of automorphisms of a simplicial complex

1988 ◽  
Vol 30 (3) ◽  
pp. 331-337 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Simplicial Complex ◽  
Elementary Proof ◽  
Universal Covering ◽  
Covering Space ◽  
Covering Spaces ◽  
Group Of Automorphisms ◽  
Generators And Relations ◽  
Universal Covering Space ◽  
Simply Connected ◽  
Special Case

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

scholarly journals On ergodic foliations

1988 ◽  
Vol 8 (3) ◽  
pp. 437-457 ◽  
Author(s):  
Kyewon Park
Keyword(s):  
Quotient Space ◽  
Group Actions ◽  
Covering Space ◽  
Ergodic Action ◽  
Covering Spaces

AbstractWe define an ergodic ℤ-foliation and show that it can be realized as a quotient space of the ‘covering space’. The covering space has two actions, T and S, where T is a ℤ-action, S is a map of order two, and S and T skew-commute; that is, STS = T−1. We study the isometry between two foliations via the isomorphism between two bigger group actions in the covering spaces. Properties of an ergodic foliation are studied in a way similar to the study of an ergodic action. We construct a counterexample of a K-automorphism to show that, unlike Bernoulli automorphisms, ℤ-actions do not completely determine ℤ-foliations.

scholarly journals Amenability, connected components, and definable actions

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Ehud Hrushovski ◽  
Krzysztof Krupiński ◽  
Anand Pillay
Keyword(s):  
Compact Space ◽  
Invariant Probability Measure ◽  
Connected Components ◽  
Topological Groups ◽  
Bohr Compactification ◽  
Weakly Almost Periodic ◽  
Closed Type ◽  
Definable Groups ◽  
The Relationship ◽  
Saturated Structure

AbstractWe study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

scholarly journals A study of some aspects of topological groups

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 55-65
Author(s):  
M.R. Adhikari ◽  
M. Rahaman
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
Base Point ◽  
Homotopy Category ◽  
Topological Spaces ◽  
Topological Groups ◽  
Homotopy Classes ◽  
Continuous Maps

The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set [X,Y], of homotopy classes of maps from a space X to a given space Y for all X which is natural with respect to X. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contra variant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphism.

Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

scholarly journals Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2382
Author(s):  
Susmit Bagchi
Keyword(s):  
Multiplicative Group ◽  
Equivalence Classes ◽  
Fundamental Groups ◽  
Local Homeomorphism ◽  
Covering Spaces ◽  
Trivial Group ◽  
Algebraic Forms ◽  
Simply Connected ◽  
Path Homotopy ◽  
Group Structures

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

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