CHAPTER I Braid Groups In this introductory chapter, we briefly explain how the groups Bn arise in several contexts of geometry and algebra, and we mention a few basic results that will be frequently used in the sequel. Our purpose here is not to be exhaustive, and many possible approaches are not mentioned—for instance the connection with con- figuration spaces. All results in this chapter are classical, and we refer to textbooks for most of the proofs see for instance [123], [118], [14], or [176]. The organization is as follows. In Section 1, we start with the Artin presentation of the group Bn in terms of generators and relations. In Section 2, we describe the connection with the geometric viewpoint of isotopy classes of families of intertwining strands. In Section 3, we address the braid group as the mapping class group of a punctured disk. Finally, in Section 4, we introduce the monoid of positive braids and mention some basic results from Garside’s theory. 1. The Artin presentation Here, we introduce the braid group Bn using the abstract presentation already mentioned in the Introduction, due to E. Artin [4]. 1.1. Braid relations. Braid groups can be specified using a standard presen- tation. Definition 1.1. For n 2, the n-strand braid group Bn is defined to be the presented group (1.1) σ1, ... , σn−1 σiσj = σjσi for |i j| 2 σ i σ j σ i = σ j σ i σ j for |i j| = 1 . The elements of Bn are called n-strand braids. The braid group on infinitely many strands, denoted B∞, is defined by a presentation with infinitely many generators σ 1 , σ 2 ,... subject to the same relations. Clearly, the identity mapping on {σ1, ... , σn−1} extends into a homomorphism of Bn to Bn+1. It can be proved easily—and it will be clear from the geometric interpretation of Section 2—that this homomorphism is injective, and, therefore, we can identify Bn with the subgroup of B∞ generated by σ 1 , ... , σ n−1 . This is the point of view we shall always adopt in the sequel. 1.2. Braid words. According to Definition 1.1, every braid admits decom- positions in terms of the generators σi and their inverses. A word on the let- ters σ±1, 1 ... , σ±1 n−1 is called an n-strand braid word. The length of a braid word w is denoted by (w). If the braid β is the equivalence class of the braid word w, we say that w represents β, or is an expression of β, and we write β = w. We say that two braid words are equivalent if they represent the same braid, i.e., if they are 9
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