scholarly journals EVERY COUNTABLE GROUP IS THE FUNDAMENTAL GROUP OF SOME COMPACT SUBSPACE OF

2015 ◽  
Vol 92 (1) ◽  
pp. 145-148
Author(s):  
ADAM J. PRZEŹDZIECKI
Keyword(s):  
Fundamental Group ◽  
Countable Group ◽  
Compact Subspace ◽  
The One ◽  
Path Connected

For every countable group $G$ we construct a compact path connected subspace $K$ of $\mathbb{R}^{4}$ such that ${\it\pi}_{1}(K)\cong G$. Our construction is much simpler than the one found recently by Virk.

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 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.

THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋

2008 ◽  
Vol 18 (08) ◽  
pp. 1259-1282 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Generic Projection ◽  
Galois Cover ◽  
The One ◽  
First Time

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

scholarly journals A fundamental group topology

1984 ◽  
Vol 37 (3) ◽  
pp. 416-420
Author(s):  
C. H. Houghton
Keyword(s):  
Topological Group ◽  
Fundamental Group ◽  
Group Topology ◽  
The One ◽  
One Point Compactification

AbstractThe topology of the Čech fundamental group of the one-point compactification of an appropriate space Y induces a topology on the fundamental group of Y. We describe this topology in terms of a topological group introduced by Higman.

scholarly journals Sur l'existence du sch\'ema en groupes fondametal

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Marco Antei ◽  
Michel Emsalem ◽  
Carlo Gasbarri
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Group Scheme ◽  
Final Version ◽  
Finite Fundamental Group ◽  
Galois Covers ◽  
The One

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group. Comment: in French. Final version (finally!)

Lifting homotopies through fixed points

1982 ◽  
Vol 93 (1-2) ◽  
pp. 123-128 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Fundamental Group ◽  
Fixed Points ◽  
Orbit Space ◽  
Connected Space ◽  
Discontinuous Group ◽  
Covering Map ◽  
Group Of Homeomorphisms ◽  
Path Connected ◽  
General Conditions

SynopsisIf G is a discontinuous group of homeomorphisms of a connected, locally path connected space X, which acts freely on X, then the projection π: X → X/G is a covering map and has the homotopy lifting property. Here we allow the elements of G to have fixed points and use work of Rhodes to investigate how two loops in X are related if their projections are homotopic in X/G. This enables us to establish a formula for the fundamental group of the orbit space of a discontinuous group under very general conditions. Finally we show by means of an example that some restriction on the action near fixed points is needed for the formula to be valid.

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 The fundamental groupoid as a topological groupoid

1975 ◽  
Vol 19 (3) ◽  
pp. 237-244 ◽  
Author(s):  
R. Brown ◽  
G. Danesh-Naruie
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Initial Point ◽  
Covering Space ◽  
Topological Groupoid ◽  
Fundamental Groupoid ◽  
Path Connected ◽  
Totally Disconnected

Let X be a topological space. Then we may define the fundamental groupoid πX and also the quotient groupoid (πX)/N for N any wide, totally disconnected, normal subgroupoid N of πX (1). The purpose of this note is to show that if X is locally path-connected and semi-locally 1-connected, then the topology of X determines a “lifted topology” on (πX)/N so that it becomes a topological groupoid over X. With this topology the subspace which is the fibre of the initial point map ∂′: (πX)/N→X over x in X, is the usual covering space of X determined by the normal subgroup N{x} of the fundamental group π(X, x).

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.

On the construction of a covering map

2019 ◽  
Vol 26 (2) ◽  
pp. 303-309
Author(s):  
Samson Saneblidze
Keyword(s):  
Fundamental Group ◽  
Geometric Realization ◽  
Short Sequence ◽  
Simplicial Approximation ◽  
Covering Map ◽  
Maximal Tree ◽  
Simplicial Set ◽  
Combinatorial Model ◽  
Path Connected

Abstract Let {Y=\lvert X\rvert} be the geometric realization of a path-connected simplicial set X, and let {G=\pi_{1}(X)} be the fundamental group. Given a subgroup {H\subset G} , let {G/H} be the set of cosets. Using the combinatorial model {\boldsymbol{\Omega}X\to\mathbf{P}X\to X} of the path fibration {{\Omega}Y\to{P}Y\to Y} and a canonical action {\mu\colon\boldsymbol{\Omega}X\times G/H\to G/H} , we construct a covering map {G/H\to Y_{H}\to Y} as the geometric realization of the associated short sequence {G/H\to\mathbf{P}X\times_{\mu}G/H\to X} . This construction, in particular, does not use the existence of a maximal tree in X. For a 2-dimensional X and {H=\{1\}} , it can also be viewed as a simplicial approximation of a Cayley 2-complex of G.

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