scholarly journals EXTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE

2019 ◽  
Vol 71 (1) ◽  
pp. 175-196
Author(s):  
Kenta Funayoshi ◽  
Yuya Koda
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Genus 2

Abstract An automorphism $f$ of a closed orientable surface $\Sigma $ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma $ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma $ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa, and the second author plays a key role.

scholarly journals Classification of periodic transformations of an orientable surface of genus two

2021 ◽  
Vol 23 (2) ◽  
pp. 147-158
Author(s):  
Denis A. Baranov ◽  
Olga V. Pochinka
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Modular Surface ◽  
Periodic Surface ◽  
Genus 2 ◽  
Genus Two ◽  
Necessary And Sufficient ◽  
Genus One

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

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 ROOTS OF DEHN TWISTS ABOUT SEPARATING CURVES

2013 ◽  
Vol 95 (2) ◽  
pp. 266-288 ◽  
Author(s):  
KASHYAP RAJEEVSARATHY
Keyword(s):  
Compatibility Condition ◽  
Simple Formula ◽  
Orientable Surface ◽  
Dehn Twist ◽  
Data Sets ◽  
Data Set ◽  
Closed Orientable Surface ◽  
Dehn Twists

AbstractLet $C$ be a curve in a closed orientable surface $F$ of genus $g\geq 2$ that separates $F$ into subsurfaces $\widetilde {{F}_{i} } $ of genera ${g}_{i} $, for $i= 1, 2$. We study the set of roots in $\mathrm{Mod} (F)$ of the Dehn twist ${t}_{C} $ about $C$. All roots arise from pairs of ${C}_{{n}_{i} } $-actions on the $\widetilde {{F}_{i} } $, where $n= \mathrm{lcm} ({n}_{1} , {n}_{2} )$ is the degree of the root, that satisfy a certain compatibility condition. The ${C}_{{n}_{i} } $-actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of ${t}_{C} $ by compatible pairs of data sets. We use these data set pairs to classify all roots for $g= 2$ and $g= 3$. We show that there is always a root of degree at least $2{g}^{2} + 2g$, while $n\leq 4{g}^{2} + 2g$. We also give some additional applications.

scholarly journals DOUBLES OF KLEIN SURFACES

2012 ◽  
Vol 54 (3) ◽  
pp. 507-515
Author(s):  
ANTONIO F. COSTA ◽  
WENDY HALL ◽  
DAVID SINGERMAN
Keyword(s):  
Klein Bottle ◽  
Orientable Surface ◽  
Historical Note ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Algebraic Functions ◽  
Genus 2

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

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 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} .

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.

Classification of real moduli spaces over genus 2 curves

1995 ◽  
Vol 57 (2) ◽  
pp. 207-215 ◽  
Author(s):  
Shuguang Wang
Keyword(s):  
Moduli Spaces ◽  
Genus 2 ◽  
Genus 2 Curves

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.

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