scholarly journals On locally 1-connectedness of quotient spaces and its applications to fundamental groups

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 27-35
Author(s):  
Ali Pakdaman ◽  
Hamid Torabi ◽  
Behrooz Mashayekhy
Keyword(s):  
Fundamental Group ◽  
Metric Space ◽  
Pairwise Disjoint ◽  
Quotient Space ◽  
Continuous Homomorphism ◽  
Fundamental Groups ◽  
Quotient Spaces ◽  
Path Connected

Let X be a locally 1-connected metric space and A1,A2,...,An be connected, locally path connected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X/(A1,A2,..., An) obtained from X by collapsing each of the sets Ai?s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X/(A1,A2,...,An).

scholarly journals Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 500
Author(s):  
Susmit Bagchi
Keyword(s):  
Fundamental Group ◽  
Fundamental Groups ◽  
3 Dimensional ◽  
Path Component ◽  
Simply Connected ◽  
Path Connected ◽  
Real Subspace ◽  
Boundary Surfaces

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space.

scholarly journals Test map characterizations of local properties of fundamental groups

2018 ◽  
Vol 12 (01) ◽  
pp. 37-85 ◽  
Author(s):  
Jeremy Brazas ◽  
Hanspeter Fischer
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closure Operator ◽  
Fundamental Groups ◽  
Space Theory ◽  
Unified Approach ◽  
Covering Map ◽  
Unique Path ◽  
Local Properties ◽  
Path Connected

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space [Formula: see text], and a subgroup [Formula: see text], we characterize the unique path lifting property relative to [Formula: see text] in terms of a new closure operator on the [Formula: see text]-subgroup lattice that is induced by maps from a fixed “test” domain into [Formula: see text]. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

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.

INFINITARY COMMUTATIVITY AND ABELIANIZATION IN FUNDAMENTAL GROUPS

2020 ◽  
pp. 1-23
Author(s):  
JEREMY BRAZAS ◽  
PATRICK GILLESPIE
Keyword(s):  
Fundamental Group ◽  
Infinite Product ◽  
Homotopy Groups ◽  
Fundamental Groups ◽  
Peano Continua ◽  
Specker Group ◽  
The Subject ◽  
Set Of Points ◽  
Path Connected ◽  
Connected Spaces

Abstract Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$ . Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

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 On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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.

ON THE COMPLETION OF -METRIC SPACES

2018 ◽  
Vol 98 (2) ◽  
pp. 298-304 ◽  
Author(s):  
NGUYEN VAN DUNG ◽  
VO THI LE HANG
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Quotient Space ◽  
Equivalence Classes ◽  
Cauchy Sequences

Based on the metrisation of $b$-metric spaces of Paluszyński and Stempak [‘On quasi-metric and metric spaces’, Proc. Amer. Math. Soc.137(12) (2009), 4307–4312], we prove that every $b$-metric space has a completion. Our approach resolves the limitation in using the quotient space of equivalence classes of Cauchy sequences to obtain a completion of a $b$-metric space.

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.

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