scholarly journals Application Of The Seifert-Van Kampen Theorem To Certain Sums Of Topological Spaces

1970 ◽  
Vol 35 (2) ◽  
pp. 197-201
Author(s):  
Mohd. Altab Hossain
Keyword(s):  
Fundamental Group ◽  
Key Words ◽  
Commutative Diagram ◽  
Topological Spaces ◽  
Fundamental Groups

Seifert-Van Kampen theorem for the sum X + Y and external sum X ⊕Y of topological spaces is studied and the fundamental groups of these sums have been determined. Key words: Sum; External sum; Fundamental group; Commutative diagram DOI: http://dx.doi.org/10.3329/jbas.v35i2.9424 JBAS 2011; 35(2): 197-201

scholarly journals MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS

2010 ◽  
Vol 12 (01) ◽  
pp. 107-119 ◽  
Author(s):  
ARMINDO COSTA ◽  
MICHAEL FARBER
Keyword(s):  
Fundamental Group ◽  
Motion Planning ◽  
Cohomological Dimension ◽  
Upper Bounds ◽  
Topological Spaces ◽  
Topological Complexity ◽  
Fundamental Groups ◽  
Planning Algorithms

We establish sharp upper bounds for the topological complexity TC(X) of motion planning algorithms in topological spaces X such that the fundamental group is "small", i.e. when π1(X) is cyclic of order ≤ 3 or has small cohomological dimension.

scholarly journals On special subgroups of fundamental group

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1403-1429
Author(s):  
Zadeh Ayatollah ◽  
Fatemeh Ebrahimifar ◽  
Mohammad Mahmoodi
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Fundamental Groups ◽  
Infinite Cardinal ◽  
Homotopy Classes ◽  
Hawaiian Earring

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

scholarly journals Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1039
Author(s):  
Susmit Bagchi
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Algebraic Topology ◽  
Quotient Space ◽  
Group Structure ◽  
Topological Spaces ◽  
Fundamental Groups ◽  
Homotopy Classes ◽  
Topological Excitations ◽  
Homotopy Decomposition

The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

scholarly journals Fundamental Group of Simple C*-algebras with Unique Trace III

2012 ◽  
Vol 64 (3) ◽  
pp. 573-587 ◽  
Author(s):  
Norio Nawata
Keyword(s):  
Fundamental Group ◽  
Lower Semicontinuous ◽  
Classification Theorem ◽  
Fundamental Groups ◽  
C Algebras ◽  
Scalar Multiple ◽  
Unique Trace ◽  
Densely Defined

Abstract We introduce the fundamental group ℱ(A) of a simple σ-inital C*-algebra A with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple C*-algebras with Unique Trace I and II by Nawata andWatatani. Our definition in this paper makes sense for stably projectionless C*-algebras. We show that there exist separable stably projectionless C*-algebras such that their fundamental groups are equal to ℝ×+ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Homotopy Equivalence of a Cofibre Fibre Composite

1977 ◽  
Vol 29 (6) ◽  
pp. 1152-1156 ◽  
Author(s):  
Philip R. Heath
Keyword(s):  
Commutative Diagram ◽  
Fibre Composite ◽  
Topological Spaces ◽  
Homotopy Equivalence ◽  
Image Position

Consider the following commutative diagram in Top, the category of topological spacesin which j and j' are cofibrations, p and p' are (Hurewicz) fibrations and ƒ0, ƒi and ƒ2 are homotopy equivalences.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

Finiteness of Isomorphism Classes of Hyperbolic Polycurves with Prescribed Fundamental Groups

2020 ◽  
Author(s):  
Koichiro Sawada
Keyword(s):  
Fundamental Group ◽  
Profinite Group ◽  
Fundamental Groups ◽  
Isomorphism Classes

Abstract In the present paper, we show that there are at most finitely many isomorphism classes of hyperbolic polycurves (i.e., successive extensions of families of hyperbolic curves) over certain types of fields whose étale fundamental group is isomorphic to a prescribed profinite group.

scholarly journals On l-Adic Iterated Integrals, III Galois Actions on Fundamental Groups

2005 ◽  
Vol 178 ◽  
pp. 1-36 ◽  
Author(s):  
Zdzisław Wojtkowiak
Keyword(s):  
Fundamental Group ◽  
Galois Representations ◽  
Fundamental Groups ◽  
Iterated Integrals ◽  
Galois Actions

We continue to study l-adic iterated integrals introduced in the first part. We shall calculate explicitly l-adic logarithm and l-adic polylogarithms. Next we shall use these results to study Galois representations on the fundamental group of .

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