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Publisher:

Springer

Publication Date:

2008

Number of Pages:

340

Format:

Hardcover

Series:

Graduate Texts in Mathematics 247

Price:

59.95

ISBN:

9780387338415

Category:

Textbook

[Reviewed by , on ]

Scott Taylor

10/13/2008

Rarely does a mathematical object have as many different descriptions as does the braid group on *n* strands. The simplest description is simply that the group *B _{n}* is generated by elements

subject to the relations

s_{i}s_{j} = s_{j}s_{i} |
if |i – j| ≥ 2 |

s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} |
if 1 ≤ i ≤ n–2 |

I am partial to two other descriptions of the braid group *B _{n}*. First, the group

Second, the *n*-strand braid group is also the mapping class group of an *n*-punctured disc, that is, the group of homeomorphisms of the punctured disc which fix the boundary modulo the subgroup of homeomorphisms isotopic to the identity.

Details on these and several other descriptions of the braid groups are carefully provided by Kassel and Turaev’s text *Braid Groups*. As is to be expected from any text on braid groups, it highlights the relationship between braid groups and links and gives detailed proofs of many classical results. (For example, the braid groups are torsion-free.) Most of these classical results are also contained in Birman’s famous *Braids, Links, and Mapping Class Groups* (Princeton, 1975). *Braid Groups* will, therefore, be most valuable for its treatment of more recent work. A few of these recent results appealed to my interests; these are the topics that I will mention in this review.

For a long time few representations of the braid group were known. The most famous of these was the Burau representation. For *n* ≥ 6, the Burau representation of *B _{n}* was proven in 1993 to be unfaithful (i.e. non-injective) by Long and Paton. In 1999, Bigelow improved this by showing that the Burau representation of

The homological view of representations of *B _{n}* also led to a resolution of the long-standing conjecture that the braid groups are linear, that is, that there exists a faithful representation of

Using Bigelow’s methods, the authors of *Braid Groups* describe this representation in terms of an action on a homology group and prove it to be faithful. This is one of the few occurences of the term “noodle” in mathematics. Bigelow’s original proof used both forks and noodles, but (sadly) the forks turn out to be unnecessary for this tasty mathematical meal.

My other two favourite sections of *Braid Groups* are the section on Garside groups (which solves the word problem for braid groups) and the section on the orderability of the braid groups. This last section presents a construction of Dehornoy of a left-invariant total order on *B _{n}*. (A left invariant total order on a group is a total order ≤ such that if

*Braid Groups* is very well written. The proofs are detailed, clear, and complete. The topics covered are now fundamental to the study of braid groups and other groups (and algebras) related to the braid groups. (A partial list includes mapping class groups, fundamental groups of configuration spaces, Iwahori-Hecke algebras, Temperley-Lieb algebras, Garside groups, Coxeter groups, and Artin groups.)

A particular strength of the text is its pattern of introducing a general concept (such as the Garside group), proving one or two theorems, and then immediately showing the applicability of the concept to the braid groups. This text is an excellent introduction to many of these concepts.

My only complaint is that there are not enough pictures. More pictures would certainly make the text easier to read, particularly for arguments concerning diagrams.

The text is to be praised for its level of detail. This is a field where it is easy to lapse into giving frustratingly imprecise proofs. The authors in all cases have avoided doing so. There are a few occasions where they refer the reader to a paper or another text for the proof of a theorem. This is to be expected in a reasonably sized textbook about a subject as broad and deep as the study of braid groups.

The authors review certain graduate-level topics (such as fibrations and group representations), but unless one has seen them before the sections of the text that rely on those concepts are not likely to be accessible. Indeed, this book would be very challenging for anyone without a substantial algebra or topology background. For people with that background who want to understand current research in braid group related areas, *Braid Groups* is an excellent, in fact indispensable, text.

Scott Taylor is Visiting Assistant Professor of Mathematics at Colby College in Waterville, ME.

Braids and Braid Groups.- Braids, Knots, and Links.- Homological Representations of the Braid Groups.- Symmetric Groups and Iwahori-Hecke Algebras.- Representations of the Iwahori-Hecke Algebras.- Garside Monoids and Braid Monoids.- An Order on the Braid Groups.- Appendix A. Presentations of SL

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