scholarly journals Enumeration of orientable coverings of a non-orientable manifold

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Jin Ho Kwak ◽  
Alexander Mednykh ◽  
Roman Nedela
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Finitely Generated ◽  
Closed Orientable Surface ◽  
Orientable Manifold ◽  
International Audience

International audience In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

1995 ◽  
Vol 04 (02) ◽  
pp. 213-224 ◽  
Author(s):  
JONATHAN A. HILLMAN ◽  
AKIO KAWAUCHI
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Flat Embedding ◽  
Orientable Surfaces

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

scholarly journals A Note on Regular Coverings of Closed Orientable Surfaces

1961 ◽  
Vol 5 (2) ◽  
pp. 49-66 ◽  
Author(s):  
Jens Mennicke
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Finite Index ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Closed Orientable Surface ◽  
Regular Covering ◽  
Genus 2 ◽  
Image Position ◽  
Orientable Surfaces

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.Let the closed orientable surfaceFhof genushbe a covering ofF2and letand f be the fundamental groups respectively. Thenis a subgroup of f of indexn = h − 1. A covering is called regular ifis normal in f.Conversely, letbe a normal subgroup of f of finite index. Then there is a uniquely determined regular coveringFhsuch thatis the fundamental group ofFh. The coveringFhis an orientable surface. Since the indexnofin f is supposed to be finite,Fhis closed, and its genus is given byn=h − 1.The fundamental group f can be defined by.

scholarly journals ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

2020 ◽  
pp. 1-10
Author(s):  
MARK GRANT ◽  
AGATA SIENICKA
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Mapping Class ◽  
Closed Orientable Surface ◽  
Outer Action ◽  
Positive Genus ◽  
Closed Sphere ◽  
Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

scholarly journals Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons

2017 ◽  
Vol 29 (06) ◽  
pp. 1750018 ◽  
Author(s):  
Sven Bachmann
Keyword(s):  
Ground State ◽  
Entanglement Entropy ◽  
Orientable Surface ◽  
Cell Complex ◽  
Local Order ◽  
Topological Order ◽  
Cohomology Groups ◽  
Closed Orientable Surface ◽  
Ground State Degeneracy ◽  
Comprehensive Study

In this comprehensive study of Kitaev’s abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterize the elementary anyonic excitations. The homology and cohomology groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterizations of topological order.

scholarly journals Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

2015 ◽  
Vol 26 (09) ◽  
pp. 1550066 ◽  
Author(s):  
Michael Brandenbursky
Keyword(s):  
Braid Group ◽  
Infinite Family ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Invariant Metrics ◽  
Closed Orientable Surface ◽  
Infinite Dimensional ◽  
Hamiltonian Diffeomorphisms ◽  
Injective Homomorphisms ◽  
Area Preserving

Let Σg be a closed orientable surface of genus g and let Diff 0(Σg, area ) be the identity component of the group of area-preserving diffeomorphisms of Σg. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σg defines a nontrivial homogeneous quasi-morphism on the group Diff 0(Σg, area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0(Σg, area ) is infinite-dimensional. Let Ham (Σg) be the group of Hamiltonian diffeomorphisms of Σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham (Σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham (Σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham (Σg).

scholarly journals The Klein bottle group is not strongly verbally closed, though awfully close to being so

2020 ◽  
pp. 1-7
Author(s):  
Anton A. Klyachko
Keyword(s):  
Fundamental Group ◽  
Closed Subgroup ◽  
Klein Bottle ◽  
Finitely Generated ◽  
Finitely Generated Group

Abstract According to Mazhuga’s theorem, the fundamental group H of anyconnected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing H as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.

scholarly journals Abstract commensurators of surface groups

2019 ◽  
pp. 1-16
Author(s):  
Khalid Bou-Rabee ◽  
Daniel Studenmund
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Computational Methods ◽  
Normal Forms ◽  
Computer Assisted ◽  
Surface Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Concrete Objects

Let [Formula: see text] be the fundamental group of a surface of finite type and [Formula: see text] be its abstract commensurator. Then [Formula: see text] contains the solvable Baumslag–Solitar groups [Formula: see text] for any [Formula: see text]. Moreover, the Baumslag–Solitar group [Formula: see text] has an image in [Formula: see text] that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of [Formula: see text] are concrete objects amenable to computational methods. For example, we give a proof that [Formula: see text] is not residually finite without the use of normal forms of HNN extensions.

scholarly journals Singular pleated surfaces and CP1–structures

1995 ◽  
Vol 37 (2) ◽  
pp. 179-190 ◽  
Author(s):  
Ser Peow Tan
Keyword(s):  
Teichmüller Space ◽  
Universal Cover ◽  
Orientable Surface ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Distinguished Point ◽  
Totally Geodesic ◽  
Closed Orientable Surface ◽  
Hyperbolic Structures ◽  
Usual Interpretation

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

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