scholarly journals Section problems for configuration spaces of surfaces

2019 ◽  
pp. 1-29
Author(s):  
Lei Chen
Keyword(s):  
Finite Type ◽  
Braid Groups ◽  
Configuration Spaces ◽  
Easy Proof ◽  
Genus Formula

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of [Formula: see text] ordered points on a surface [Formula: see text] of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf[Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points in [Formula: see text]. Let [Formula: see text] be the map given by [Formula: see text]. We classify all continuous sections of [Formula: see text] up to homotopy by proving the following: (1) If [Formula: see text] and [Formula: see text], any section of [Formula: see text] is either “adding a point at infinity” or “adding a point near [Formula: see text]”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near [Formula: see text]” or “adding a point at infinity” depends in a delicate way on properties of [Formula: see text].) (2) If [Formula: see text] a [Formula: see text]-sphere and [Formula: see text], any section of [Formula: see text] is “adding a point near [Formula: see text]”; if [Formula: see text] and [Formula: see text], the bundle [Formula: see text] does not have a section. (We define this term in Sec. 3.2). (3) If [Formula: see text] a surface of genus [Formula: see text] and for [Formula: see text], we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle [Formula: see text] does not have a section.

Marked homeomorphisms and the realization problem

1996 ◽  
Vol 119 (4) ◽  
pp. 575-596
Author(s):  
Peter Greenberg
Keyword(s):  
Positive Response ◽  
Braid Groups ◽  
Configuration Spaces ◽  
Topological Invariants ◽  
Discrete Groups ◽  
Fundamental Groups ◽  
Realization Problem ◽  
And Topology

The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

STATISTICS AND SPIN ON TWO-DIMENSIONAL SURFACES

1991 ◽  
Vol 06 (30) ◽  
pp. 2801-2810 ◽  
Author(s):  
A. P. BALACHANDRAN ◽  
T. EINARSSON ◽  
T. R. GOVINDARAJAN ◽  
R. RAMACHANDRAN
Keyword(s):  
Braid Groups ◽  
Configuration Spaces ◽  
Fundamental Groups ◽  
Two Dimensional ◽  
Connected Sums

The fundamental groups of the configuration spaces for the O(3) nonlinear σ-model on the compact genus g surfaces [Formula: see text] and on the connected sums [Formula: see text] are known for any soliton number N. So are the braid for N spinless particles on these manifolds. The representations of these groups govern the possible statistics of solitons and particles. We show that when spin and creation/annihilation processes are introduced, the fundamental groups for the particles are the same as the corresponding σ-model groups. These fundamental groups incorporate the spin-statistics connection and are of greater physical relevance than the standard braid groups.

Braids

2017 ◽  
Author(s):  
Aaron Abrams
Keyword(s):  
Group Theory ◽  
Knot Theory ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
Configuration Spaces ◽  
Research Projects ◽  
And Topology ◽  
Mathematics And Science

This chapter deals with mathematical braids, an idealized abstraction of the familiar hair and bread braids. In a mathematical braid, the “strings” remain separate at the ends rather than being fused together. Furthermore, a mathematical braid can have any number of braided strings and the braiding can occur in any pattern. This chapter gives different ways to think about mathematical braids and some of the basic theorems about them. It also describes several ways that braids relate to other parts of mathematics and science such as robotics, knot theory, and hyperplane arrangements. After providing an overview of some group theory relating to braids, the chapter considers configuration spaces that connect braid groups and topology as well as the concept of punctured disks. Finally, it presents an experiment for braiding the hair by first making a ponytail and then doing the braiding. The discussion includes exercises and research projects.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

New Signature Scheme over the Braid Groups

2011 ◽  
Vol 32 (12) ◽  
pp. 2930-2934
Author(s):  
Yun Wei ◽  
Guo-hua Xiong ◽  
Wan-su Bao ◽  
Xing-kai Zhang
Keyword(s):  
Braid Groups ◽  
Signature Scheme
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