Chaotic Expansive Homeomorphisms on Closed Orientable Surfaces of Positive Genus

2015 ◽  
Vol 3 (3) ◽  
pp. 291-314
Author(s):  
Jiehua Mai ◽  
Song Shao
Keyword(s):  
Positive Genus ◽  
Orientable Surfaces

Gropes, Towers, and Skyscrapers

2021 ◽  
pp. 171-184
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Manifolds With Boundary ◽  
Good Group ◽  
Positive Genus ◽  
Symplectic Basis ◽  
Orientable Surfaces

Gropes, towers, and skyscrapers are carefully defined. These are the objects that the rest of Part II studies and seeks to construct. All three are 4-manifolds with boundary, obtained from stacking thickened surfaces on top of one another. Gropes are constructed from thickened orientable surfaces with positive genus, each stage attached to a symplectic basis of curves for the homology of the previous stage. Towers have an additional type of stage obtained from plumbed thickened discs. A skyscraper is the endpoint compactification of an infinite tower. An introduction to endpoint compactifications is included. The notion of a good group is also defined.

scholarly journals ON LIE BIALGEBRAS OF LOOPS ON ORIENTABLE SURFACES

2008 ◽  
Vol 17 (03) ◽  
pp. 351-359 ◽  
Author(s):  
ATTILIO LE DONNE
Keyword(s):  
Negative Answer ◽  
Orientable Surface ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Genus Zero ◽  
Homotopy Classes ◽  
Positive Genus ◽  
Simple Class ◽  
Orientable Surfaces

Goldman [2] and Turaev [4] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas [1] by the aid of the computer, found a negative answer to Turaev's question about the characterization of the classes with cobracket zero as multiples of simple classes, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if, for genus zero, every class with cobracket zero is a multiple of a simple class. The aim of this paper is to give a positive answer to this conjecture.

scholarly journals Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

2021 ◽  
pp. 2150040
Author(s):  
Vukašin Stojisavljević ◽  
Jun Zhang
Keyword(s):  
Cotangent Bundle ◽  
Isometric Embedding ◽  
Large Scale ◽  
Riemannian Metrics ◽  
Closed Geodesics ◽  
Persistence Modules ◽  
Large Scale Geometry ◽  
Nonlinear Version ◽  
Positive Genus ◽  
Orientable Surfaces

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text] The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text] Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text] On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

A Polynomial Invariant of Graphs On Orientable Surfaces

2001 ◽  
Vol 83 (3) ◽  
pp. 513-531 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Polynomial Invariant ◽  
Orientable Surfaces

Stable bi-maps from closed orientable surfaces to R x R

2021 ◽  
Author(s):  
Catarina Mendes de Jesus ◽  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa
Keyword(s):  
Orientable Surfaces

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

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