Geometry & Topology

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

Stephen Bigelow

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Abstract

The Burau representation is a natural action of the braid group Bn on the free [t,t1]–module of rank n1. 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 n9 and Long and Paton improved on Moody’s techniques to bring this down to n6. 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 n5.

Article information

Source
Geom. Topol., Volume 3, Number 1 (1999), 397-404.

Dates
Received: 21 July 1999
Accepted: 23 November 1999
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883152

Digital Object Identifier
doi:10.2140/gt.1999.3.397

Mathematical Reviews number (MathSciNet)
MR1725480

Zentralblatt MATH identifier
0942.20017

Subjects
Primary: 20F36: Braid groups; Artin groups
Secondary: 57M07: Topological methods in group theory 20C99: None of the above, but in this section

Keywords
braid group Burau representation

Citation

Bigelow, Stephen. The Burau representation is not faithful for $n = 5$. Geom. Topol. 3 (1999), no. 1, 397--404. doi:10.2140/gt.1999.3.397. https://projecteuclid.org/euclid.gt/1513883152


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References

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  • John Atwell Moody, The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Amer. Math. Soc. 25 (1991) 379–384