scholarly journals On ergodic foliations

1988 ◽  
Vol 8 (3) ◽  
pp. 437-457 ◽  
Author(s):  
Kyewon Park
Keyword(s):  
Quotient Space ◽  
Group Actions ◽  
Covering Space ◽  
Ergodic Action ◽  
Covering Spaces

AbstractWe define an ergodic ℤ-foliation and show that it can be realized as a quotient space of the ‘covering space’. The covering space has two actions, T and S, where T is a ℤ-action, S is a map of order two, and S and T skew-commute; that is, STS = T−1. We study the isometry between two foliations via the isomorphism between two bigger group actions in the covering spaces. Properties of an ergodic foliation are studied in a way similar to the study of an ergodic action. We construct a counterexample of a K-automorphism to show that, unlike Bernoulli automorphisms, ℤ-actions do not completely determine ℤ-foliations.

scholarly journals Limits of covering spaces and residual properties of groups

1994 ◽  
Vol 36 (3) ◽  
pp. 277-282
Author(s):  
Jon Michael Corson
Keyword(s):  
Covering Space ◽  
Covering Spaces ◽  
Residually Finite ◽  
Residual Properties ◽  
Space Approach

The purpose of this paper is to point out a flaw in H. B. Griffiths' covering space approach to residual properties of groups [3]. One is led to this paper from Lyndon and Schupp's book [4, pp. 114, 141] where it is cited for covering space methods and a proof that F-groups are residually finite. However the main result of [3] is false.

scholarly journals A presentation for a group of automorphisms of a simplicial complex

1988 ◽  
Vol 30 (3) ◽  
pp. 331-337 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Simplicial Complex ◽  
Elementary Proof ◽  
Universal Covering ◽  
Covering Space ◽  
Covering Spaces ◽  
Group Of Automorphisms ◽  
Generators And Relations ◽  
Universal Covering Space ◽  
Simply Connected ◽  
Special Case

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

Orbifold Morse–Smale–Witten complexes

2014 ◽  
Vol 25 (05) ◽  
pp. 1450040 ◽  
Author(s):  
Cheol-Hyun Cho ◽  
Hansol Hong
Keyword(s):  
Critical Points ◽  
Quotient Space ◽  
Group Actions ◽  
The Self ◽  
Gradient Flows ◽  
Weak Group ◽  
Singular Homology ◽  
Invariant Part ◽  
Unstable Manifolds ◽  
Morse Complex

Given a Morse–Smale function on an effective orientable orbifold, we construct its Morse–Smale–Witten complex. We show that critical points of a certain type have to be discarded to build a complex properly, and that gradient flows should be counted with suitable weights. Its homology is proven to be isomorphic to the singular homology of the quotient space under the self-indexing assumption. For a global quotient orbifold [M/G], such a complex can be understood as the G-invariant part of the Morse complex of M, where the G-action on generators of the Morse complex has to be defined including orientation spaces of unstable manifolds at the critical points. Alternatively in the case of global quotients, we introduce the notion of weak group actions on Morse–Smale–Witten complexes for non-invariant Morse–Smale functions on M, which give rise to genuine group actions on the level of homology.

Harmonic manifolds and embedded surfaces arising from a super regular tesselation

2017 ◽  
Vol 26 (09) ◽  
pp. 1743003
Author(s):  
G. Brumfiel ◽  
H. Hilden ◽  
M. T. Lozano ◽  
J. M. Montesinos ◽  
E. Ramirez ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Quotient Space ◽  
Hyperbolic Manifolds ◽  
Covering Space ◽  
Hyperbolic Surfaces ◽  
Harmonic Manifolds ◽  
Embedded Surfaces

The main result of this paper is the construction of two Hyperbolic manifolds, [Formula: see text] and [Formula: see text], with several remarkable properties: (1) Every closed orientable [Formula: see text]-manifold is homeomorphic to the quotient space of the action of a group of order [Formula: see text] on some covering space of [Formula: see text] or [Formula: see text]. (2) [Formula: see text] and [Formula: see text] are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way. (3) There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic [Formula: see text] each of which is tesselated by regular right angled pentagons and embedded in [Formula: see text] or [Formula: see text]. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic [Formula: see text].

scholarly journals A note on the variation of Riemann surfaces

1996 ◽  
Vol 142 ◽  
pp. 1-4 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Riemann Surface ◽  
Complex Plane ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Universal Covering ◽  
Covering Space ◽  
Uniformization Theorem ◽  
Covering Spaces ◽  
Universal Covering Space

Let X be any Riemann surface. By Koebe’s uniformization theorem we know that the universal covering space of X is conformally equivalent to either Riemann sphere, complex plane, or the unit disc in the complex plane. If X is allowed to vary with parameters we may inquire the parameter dependence of the corresponding family of the universal covering spaces.

scholarly journals Real algebraic links in S3 and braid group actions on the set of n-adic integers

2020 ◽  
Vol 29 (06) ◽  
pp. 2050039
Author(s):  
Benjamin Bode
Keyword(s):  
Configuration Space ◽  
Complex Plane ◽  
Braid Group ◽  
Infinite Sequence ◽  
Group Actions ◽  
Infinite Family ◽  
Natural Numbers ◽  
Isolated Singularities ◽  
Covering Spaces ◽  
Real Polynomials

We construct an infinite tower of covering spaces over the configuration space of [Formula: see text] distinct nonzero points in the complex plane. This results in an action of the braid group [Formula: see text] on the set of [Formula: see text]-adic integers [Formula: see text] for all natural numbers [Formula: see text]. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid [Formula: see text] an infinite sequence of braids, whose braid types are invariants of [Formula: see text]. We present computations for the cases of [Formula: see text] and [Formula: see text] and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials [Formula: see text].

scholarly journals FINITELY DOMINATED COVERING SPACES OF 3- AND 4-MANIFOLDS

2008 ◽  
Vol 84 (1) ◽  
pp. 99-108
Author(s):  
JONATHAN A. HILLMAN
Keyword(s):  
Surface Group ◽  
Covering Space ◽  
Infinite Index ◽  
Covering Spaces ◽  
Ascendant Subgroup ◽  
Torsion Free

AbstractIf P is a closed 3-manifold the covering space associated to a finitely presentable subgroup ν of infinite index in π1(P) is finitely dominated if and only if P is aspherical or $\raisebox {11pt}{}\widetilde {P}\simeq {S^2}$. There is a corresponding result in dimension 4, under further hypotheses on π and ν. In particular, if M is a closed 4-manifold, ν is an ascendant, FP3, finitely-ended subgroup of infinite index in π1(M), π is virtually torsion free and the associated covering space is finitely dominated then either M is aspherical or $\widetilde {M}\simeq {S^2}$ or S3. In the aspherical case such an ascendant subgroup is usually Z, a surface group or a PD3-group.

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