Braids

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

This is an introduction to the theory of braids, one of the most beautiful examples of algebra and topology working hand-in-hand. The emphasis is on the variety of points of view from which one can view braids, and the information one gains from each aspect. As preparation for further lectures in a summer school and conference on the braid groups, these talks (and therefore this chapter) are intended for students with some basic background in topology and the theory of groups.

We discuss simplicial objects and homotopy groups. The first section covers the topics on Δ-objects and homology. The relations between Δ-sets, simplicial complexes and Δ-complexes are given. The simplicial and singular homology can be directly obtained as the derived functors of Δ-sets. The second section covers the topics on simplicial sets and homotopy. After introducing the definition of simplicial sets, the relations between simplicial sets, Δ-sets and simplicial complexes are discussed. Then, we give the connections between simplicial sets and spaces. By introducing fibrant simplicial sets, the homotopy theory on the category of simplicial sets can be set up. In particular, the homotopy groups can be combinatorially defined using simplicial sets. In the last section, we discuss simplicial group theory. The Moore chain complexes are deeply discussed. As an important example, the general homotopy groups of the 2-sphere can be described as the center of certain combinatorially given groups. We give new approaches to many classical results. Also there are many new results on these topics.

The following sections are included:

• Introduction

• Basic Definitions

• Fibrations

• On Cross-sections for Configuration Spaces

• Preparation for Theorems 3.2 and 3.3

• Proof of Theorem 3.2

• Proof of Theorem 3.3

• Surfaces, Braid Groups and Connections to Mapping Class Groups

• On Configurations in S²

• On Configurations in S¹ × S¹

• On Configurations in a Surface of Genus Greater than 1

• Loop Spaces of Configuration Spaces

• Planetary Motion in Configuration Spaces

• Homological Calculations for ℝ^m

• On Spheres Embedded in the Configuration Space

• Simplicial Objects, and Δ-Objects

• Pure Braid Groups, and Vassiliev Invariants

• On Θ_n

• Pure Braid Groups of Surfaces as Simplicial Groups and Δ-Groups

• Brunnian Braids, “Almost Brunnian” Braids, and Homotopy Groups

• Other Connections

• Questions

• Appendix. Brunnian Braids and Principal Congruence Subgroups

• Acknowledgments

• References

Braids are intimately related to configuration spaces of points. These configuration spaces give a useful model of autonomous agents (or robots) in an environment. Problems of relevance to autonomous engineering systems (e.g., motion planning, coordination, cooperation, assembly) are directly related to topological and geometric properties of configuration spaces, including their braid groups. This chapter details this correspondence, and explore several novel examples of configuration spaces relevant to applications in robotics. No familiarity with robotics is assumed.

Magnetic field lines can exhibit a wide range of topological structures, including braid structure. This chapter describes the importance of braided magnetic fields in the atmosphere of the sun. Braided coronal loops store vast amounts of energy; this energy can power small flares and heat the solar atmosphere to millions of degrees. A simple model for self-organization of braided loops will be presented. The basic theory of magnetized fluids will be briefly reviewed, emphasizing topological quantities.

In the last decade, a number of public-key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them.

This survey includes: Basic facts on braid groups and on Garside's normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of some cryptosystems which are based on other non-commutative groups).