The Burau representation is not faithful for n = 5

Bigelow, Stephen¹

¹ Department of Mathematics, University of California at Berkeley, Berkeley, California 94720, USA

Geometry & topology, Tome 3 (1999) no. 1, pp. 397-404.

Résumé

The Burau representation is a natural action of the braid group $B_n$ on the free $\mathbb{Z}\left[t,t^{-1}\right]$–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 $n\ge 9$ and Long and Paton improved on Moody’s techniques to bring this down to $n\ge 6$. 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 $n\ge 5$.

Détail

DOI : 10.2140/gt.1999.3.397

Keywords: braid group, Burau representation

Bibliographie

[1] J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)

[2] , Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284

[3] D D Long, M Paton, The Burau representation is not faithful for $n\geq 6$, Topology 32 (1993) 439

[4] J A Moody, The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Amer. Math. Soc. $($N.S.$)$ 25 (1991) 379