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.

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 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.

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions

1992 ◽  
Vol 120 (1-2) ◽  
pp. 47-60 ◽  
Author(s):  
J. W. Rutter
Keyword(s):  
Fundamental Group ◽  
Unit Group ◽  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Homotopy Equivalence ◽  
Special Cases ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.

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 Almost-continuous path connected spaces

1981 ◽  
Vol 4 (4) ◽  
pp. 823-825
Author(s):  
Larry L. Herrington ◽  
Paul E. Long
Keyword(s):  
Continuous Function ◽  
Connected Components ◽  
Continuous Path ◽  
Open Set ◽  
Regular Open ◽  
Path Connected ◽  
Almost Continuous ◽  
Connected Spaces

M. K. Singal and Asha Rani Singal have defined an almost-continuous functionf:X→Yto be one in which for eachx∈Xand each regular-open setVcontainingf(x), there exists an openUcontainingxsuch thatf(U)⊂V. A spaceYmay now be defined to be almost-continuous path connected if for eachy0,y1∈Ythere exists an almost-continuousf:I→Ysuch thatf(0)=y0andf(1)=y1An investigation of these spaces is made culminating in a theorem showing when the almost-continuous path connected components coincide with the usual components ofY.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document