scholarly journals On the modulus of the free homotopy class of a simple closed curve on an arbitrary Riemann surface

1986 ◽  
Vol 12 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Masahiko TANTGUCHI
Keyword(s):  
Riemann Surface ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Free Homotopy Class

scholarly journals Representing Homology Classes on Surfaces

1976 ◽  
Vol 19 (3) ◽  
pp. 373-374 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Unit Circle ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

scholarly journals ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS

2014 ◽  
Vol 56 (3) ◽  
pp. 681-689
Author(s):  
ZONGLIANG SUN
Keyword(s):  
Quasiconformal Mapping ◽  
Homotopy Class ◽  
Quadratic Differential ◽  
Riemann Surfaces ◽  
Simple Closed Curve ◽  
Quasiconformal Mappings ◽  
Closed Curve

AbstractIn this paper, we consider the question about length distortions under quasiconformal mappings with respect to quadratic differential metrics. More precisely, let X and Y be closed Riemann surfaces with genus at least 2, and f: X → Y being a K-quasiconformal mapping. Given two quadratic differential metrics |q1| and |q2| with unit areas on X and Y respectively, whether there exists a constant $\mathcal C$ depending only on K such that $\frac{1}{\mathcal C} l_{q_1} (\gamma) \leq l_{q_2} (f(\gamma)) \leq \mathcal C l_{q_1} (\gamma)$ holds for any simple closed curve γ ⊂ X. Here lqi(α) denotes the infimum of the lengths of curves in the homotopy class of α with respect to the metric |qi|, i = 1, 2. We give positive answers to this question, including the aspects that the desired constant ${\mathcal C}$ explicitly depends on q1, q2 and K, and that the constant $\mathcal C$ is universal for all the quantities involved.

scholarly journals A combinatorial algorithm for visualizing representatives with minimal self-intersection

2015 ◽  
Vol 24 (11) ◽  
pp. 1550058 ◽  
Author(s):  
Chris Arettines
Keyword(s):  
Homotopy Class ◽  
Orientable Surface ◽  
Closed Curve ◽  
Combinatorial Algorithm ◽  
Minimal Self ◽  
Free Homotopy Class

Given an orientable surface with boundary and a free homotopy class of a closed curve on this surface, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self-intersection.

A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

1960 ◽  
Vol 12 ◽  
pp. 209-230 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
History Of ◽  
Homogeneous Plane ◽  
Plane Continuum

One of the unsolved problems of plane topology is the following:Question. What are the homogeneous bounded plane continua?A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

scholarly journals Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

scholarly journals Handel’s fixed point theorem revisited

2012 ◽  
Vol 33 (5) ◽  
pp. 1584-1610
Author(s):  
JULIANA XAVIER
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unit Disk ◽  
Point Theorem ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Disk ◽  
Oriented Cycle

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

A Census of Planar Triangulations

1962 ◽  
Vol 14 ◽  
pp. 21-38 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Finite Number ◽  
Interior Point ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Region ◽  
Straight Segments

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.

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