International Congress of Mathematicians

ICM 2002 • Vol. II • 37-45Representations of Braid GroupsS. Bigelow*AbstractIn this paper we survey some work on representations of B n given by theinduced action on a homology module of some space. One of these, called theLawrence-Krammer representation, recently came to prominence when it wasshown to be faithful for all n. We will outline the methods used, applying themto a closely related representation for which the proof is slightly easier. Themain tool is the Blanchfield pairing, a sesquilinear pairing between elementsof relative homology. We discuss two other applications of the Blanchfieldpairing, namely a proof that the Burau representation is not faithful for largen, and a homological definition of the Jones polynomial. Finally, we discusspossible applications to the representation theory of the Hecke algebra, andultimately of the symmetric group over fields of non-zero characteristic.2000 Mathematics Subject Classification: 20F36, 20C08.Keywords and Phrases: Braid groups, Configuration spaces, Homologicalrepresentations, Blanchfield pairing.1. IntroductionArtin's braid group B n was originally defined as a group of geometric braidsin R 3 . Representations of B n have been studied for their own intrinsic interest, andalso in connection to other areas of mathematics, most notably to knot invariantssuch as the Jones polynomial.We will use the definition of B n as the mapping class group of an n-timespunctured disk D n . A rich source of representations of B n is the induced action onhomology modules of spaces related to D n . The Burau representation, one of thesimplest and best known representations of braid groups, is most naturally definedas the induced action of B n on the first homology of a cyclic covering space of D n .Lawrence [9] extended this idea to configuration spaces in D n , and was able toobtain all of the so-called Temperley-Lieb representations.* Department of Mathematics & Statistics, University of Melbourne, Victoria 3010, Australia.E-mail: bigelow@unimelb.edu.au