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The Burau representation is not faithful for $n = 5$

Stephen Bigelow

Geometry & Topology 3 (1999) 397–404

arXiv: math.GT/9904100

Abstract

The Burau representation is a natural action of the braid group Bn on the free [t,t1]–module of rank n 1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n = 3. Moody has shown that it is not faithful for n 9 and Long and Paton improved on Moody’s techniques to bring this down to n 6. Their construction uses a simple closed curve on the 6–punctured disc with certain homological properties. In this paper we give such a curve on the 5–punctured disc, thus proving that the Burau representation is not faithful for n 5.

Keywords
braid group, Burau representation
Mathematical Subject Classification
Primary: 20F36
Secondary: 57M07, 20C99
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Publication
Received: 21 July 1999
Accepted: 23 November 1999
Published: 30 November 1999
Proposed: Joan Birman
Seconded: Shigeyuki Morita, Dieter Kotschick
Authors
Stephen Bigelow
Department of Mathematics
University of California at Berkeley
Berkeley
California 94720
USA