scholarly journals Unique pseudolifting property in digital topology

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 739-746
Author(s):  
Sang-Eon Han
Keyword(s):  
Homotopy Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Digital Version ◽  
Digital Covering ◽  
Digital Spaces

The generalized universal lifting property plays an important role in classical topology. In digital topology we also have its digital version [5, 6, 14]. More precisely, the paper [6] established the concept of a digital covering (see also [11, 17]). It has substantially contributed to the calculation of digital fundamental groups of some digital spaces, the classification of digital spaces and so forth. The paper [6] also established the unique lifting property of a digital covering which plays an important role in studying both digital covering and digital homotopy theory. Motivated by the unique lifting property, the paper develops a pseudocovering which is weaker than a digital covering and investigates its various properties. Furthermore, the paper proves that a pseudocovering with some hypothesis has the unique pseudolifting property which is weaker than the unique lifting property in digital covering theory.

scholarly journals Ultra regular covering space and its automorphism group

2010 ◽  
Vol 20 (4) ◽  
pp. 699-710 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Automorphism Group ◽  
Transformation Group ◽  
Covering Space ◽  
Fundamental Groups ◽  
Discrete Transformation ◽  
Local Isomorphism ◽  
Digital Space ◽  
Regular Covering ◽  
Digital Covering ◽  
Digital Spaces

Ultra regular covering space and its automorphism groupIn order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck's discrete transformation group) of a digital covering. By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008; Han, 2006b; 2008b; 2008d; 2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open. In order to examine this problem, the present paper establishes the notion of an ultra regular covering space, studies its various properties and calculates an automorphism group of the ultra regular covering space. In particular, the paper develops the notion of compatible adjacency of a digital wedge. By comparing an ultra regular covering space with a regular covering space, we can propose strong merits of the former.

scholarly journals Non-ultra regular digital covering spaces with nontrivial automorphism groups

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1205-1218 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Automorphism Group ◽  
Homotopy Theory ◽  
Automorphism Groups ◽  
Digital Images ◽  
Covering Space ◽  
Transformation Groups ◽  
Covering Spaces ◽  
Nontrivial Automorphism ◽  
Digital Covering

The study of digital covering transformation groups (or automorphism groups, discrete deck transformation groups) plays an important role in the classification of digital spaces (or digital images). In particular, the research into transitive or nontransitive actions of automorphism groups of digital covering spaces is one of the most important issues in digital covering and digital homotopy theory. The paper deals with the problem: Is there a digital covering space which is not ultra regular and has an automorphism group which is not trivial? To solve the problem, let us consider a digital wedge of two simple closed ki-curves with a compatible adjacency, i ?{1,2}, denoted by (X, k). Since the digital wedge (X, k) has both infinite or finite fold digital covering spaces, in the present paper some of these infinite fold digital covering spaces were found not to be ultra regular and further, their automorphism groups are not trivial, which answers the problem posed above. These findings can be substantially used in classifying digital covering spaces and digital images so that the paper improves on the research in Section 4 of [3] (compare Figure 2 of the present paper with Figure 2 of [3]), which corrects an error that appears in the Boxer and Karaca's paper [3] (see the points (0,0), (0,8), (6, -1) and (6,7) in Figure 2 of [3]).

scholarly journals Equivalent conditions for digital covering maps

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4005-4014
Author(s):  
Ali Pakdaman ◽  
Mehdi Zakki
Keyword(s):  
Local Isomorphism ◽  
Covering Map ◽  
Unique Path ◽  
Continuous Surjection ◽  
Digital Covering ◽  
Equivalent Conditions ◽  
Digital Spaces

It is known that every digital covering map p:(E,k) ? (B,?) has the unique path lifting property. In this paper, we show that its inverse is true when the continuous surjective map p has no conciliator point. Also, we prove that a digital (k,?)-continuous surjection p:(E,k)? (B,?) is a digital covering map if and only if it is a local isomorphism, when all digital spaces are connected. Moreover, we find out a loop criterion for a digital covering map to be a radius n covering map.

scholarly journals Digital Topological Properties of an Alignment of Fixed Point Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 921 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Unsolved Problem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Topological Properties ◽  
Point Sets ◽  
Fixed Point Sets ◽  
Connected Spaces

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image ( X , k ) , let F ( X ) be the set of cardinalities of the fixed point sets of all k-continuous self-maps of ( X , k ) (see Definition 4). In this paper we call it an alignment of fixed point sets of ( X , k ) . Then we have the following unsolved problem. How many components are there in F ( X ) up to 2-connectedness? In particular, let C k n , l be a simple closed k-curve with l elements in Z n and X : = C k n , l 1 ∨ C k n , l 2 be a digital wedge of C k n , l 1 and C k n , l 2 in Z n . Then we need to explore both the number of components of F ( X ) up to digital 2-connectivity (see Definition 4) and perfectness of F ( X ) (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models C 2 n n , 4 , C 3 n − 1 n , 4 , and C k n , 6 play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that X ♯ ≥ 2 .

scholarly journals On the classification of rank-two representations of quasiprojective fundamental groups

2008 ◽  
Vol 144 (5) ◽  
pp. 1271-1331 ◽  
Author(s):  
Kevin Corlette ◽  
Carlos Simpson
Keyword(s):  
Fundamental Groups ◽  
Monodromy At Infinity ◽  
Dense Representation

AbstractSuppose that X is a smooth quasiprojective variety over ℂ and ρ:π1(X,x)→SL(2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X→Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.

scholarly journals Classification of the fundamental groups of join-type curves of degree seven

2015 ◽  
Vol 67 (2) ◽  
pp. 663-698
Author(s):  
Christophe EYRAL ◽  
Mutsuo OKA
Keyword(s):  
Fundamental Groups ◽  
Type Curves

scholarly journals GROUPS OF LOCALLY-FLAT DISK KNOTS AND NON-LOCALLY-FLAT SPHERE KNOTS

2005 ◽  
Vol 14 (02) ◽  
pp. 189-215 ◽  
Author(s):  
GREG FRIEDMAN
Keyword(s):  
Fundamental Group ◽  
Piecewise Linear ◽  
Complete Classification ◽  
Fundamental Groups ◽  
Singular Set ◽  
Flat Disk ◽  
Set Of Points

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

scholarly journals Classification of negatively pinched manifolds with amenable fundamental groups

2006 ◽  
Vol 196 (2) ◽  
pp. 229-260 ◽  
Author(s):  
Igor Belegradek ◽  
Vitali Kapovitch
Keyword(s):  
Fundamental Groups

On the Algebra of Semigroup Diagrams

1997 ◽  
Vol 07 (03) ◽  
pp. 313-338 ◽  
Author(s):  
Vesna Kilibarda
Keyword(s):  
Homotopy Theory ◽  
Structural Information ◽  
Geometric Method ◽  
Fundamental Groups ◽  
Free Products ◽  
Semigroup Presentation ◽  
Fundamental Groupoid ◽  
Low Dimensional ◽  
Semigroup Diagrams ◽  
Natural Way

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

Sign in / Sign up

Export Citation Format

Share Document