scholarly journals Covering groupoids of categorical rings

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Osman Mucuk ◽  
Serap Demir
Keyword(s):  
Topological Group ◽  
Covering Spaces ◽  
Categorical Group ◽  
Fundamental Groupoid ◽  
Group Object ◽  
Literature Group ◽  
Simply Connected ◽  
Path Connected ◽  
Covering Groups

A categorical group is a kind of categorization of group and similarly a categorical ring is a categorization of ring. For a topological group X, the fundamental groupoid ?X is a group object in the category of groupoids, which is also called in literature group-groupoid or 2-group. If X is a path connected topological group which has a simply connected cover, then the category of covering groups of X and the category of covering groupoids of ?X are equivalent. Recently it was proved that if (X, x0) is an H-group, then the fundamental groupoid ?X is a categorical group and the category of the covering spaces of (X, x0) is equivalent to the category of covering groupoids of the categorical group ?X. The purpose of this paper is to present similar results for rings and categorical rings.

scholarly journals Covering groups of non-connected topological groups revisited

1994 ◽  
Vol 115 (1) ◽  
pp. 97-110 ◽  
Author(s):  
Ronald Brown ◽  
Osman Mucuk
Keyword(s):  
Topological Group ◽  
Universal Cover ◽  
Topological Groups ◽  
Covering Spaces ◽  
Unique Structure ◽  
Underlying Space ◽  
Path Connected ◽  
Covering Groups

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in with pẽ = e there is a unique structure of topological group on such that ẽ is the identity and is a morphism of groups. We say that the structure of topological group on X lifts to .

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

scholarly journals SPIKES IN COSMIC CRYSTALLOGRAPHY

2002 ◽  
Vol 11 (06) ◽  
pp. 869-891 ◽  
Author(s):  
G. I. GOMERO ◽  
A. F. F. TEIXEIRA ◽  
M. J. REBOUÇCAS ◽  
A. BERNUI
Keyword(s):  
Multiple Images ◽  
Multiply Connected ◽  
Major Consequence ◽  
Statistical Fluctuations ◽  
Covering Group ◽  
Connected Case ◽  
Pair Separation ◽  
Simply Connected ◽  
The Universe ◽  
Covering Groups

If the universe is multiply connected and small the sky shows multiple images of cosmic objects, correlated by the covering group of the 3-manifold used to model it. These correlations were originally thought to manifest as spikes in pair separation histograms (PSH) built from suitable catalogues. Using probability theory we derive an expression for the expected pair separation histogram (EPSH) in a rather general topological-geometrical-observational setting. As a major consequence we show that the spikes of topological origin in PSH's are due to translations, whereas other isometries manifest as tiny deformations of the PSH corresponding to the simply connected case. This result holds for all Robertson–Walker spacetimes and gives rise to two basic corollaries: (i) that PSH's of Euclidean manifolds that have the same translations in their covering groups exhibit identical spike spectra of topological origin, making clear that even if the universe is flat the topological spikes alone are not sufficient for determining its topology; and (ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological origin. These corollaries ensure that cosmic crystallography, as originally formulated, is not a conclusive method for unveiling the shape of the universe. We also present a method that reduces the statistical fluctuations in PSH's built from simulated catalogues.

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 Topologized Soft Fundamental Group of Soft Topological Group

2020 ◽  
pp. 422-427
Author(s):  
Hiyam Hassan Kadhem ◽  
Noor Abdul Moneem Jawad
Keyword(s):  
Topological Group ◽  
Fundamental Group ◽  
Identity Element ◽  
Connected Space ◽  
Path Connected

      In this paper, we show that each soft topological group is a strong small soft loop transfer space at the identity element. This indicates that the soft quasitopological fundamental group of a soft connected and locally soft path connected space, is a soft topological group.

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.

scholarly journals An example of a non-Borel locally-connected finite-dimensional topological group

2017 ◽  
Vol 9 (1) ◽  
pp. 3-5
Author(s):  
I.Ya. Banakh ◽  
T.O. Banakh ◽  
M.I. Vovk
Keyword(s):  
Topological Group ◽  
Lie Group ◽  
Classical Theorem ◽  
Locally Compact ◽  
Local Connectedness ◽  
Connected Subgroup ◽  
Finite Dimensional ◽  
Path Connectedness ◽  
Local Path ◽  
Path Connected

According to a classical theorem of Gleason and Montgomery, every finite-dimensional locally path-connected topological group is a Lie group. In the paper for every $n\ge 2$ we construct a locally connected subgroup $G\subset{\mathbb R}^{n+1}$ of dimension $\dim(G)=n$, which is not locally compact. This answers a question posed by S. Maillot on MathOverflow and shows that the local path-connectedness in the result of Gleason and Montgomery can not be weakened to the local connectedness.

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