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In [[mathematics]] the '''Burau representation''' is a [[group representation|representation]] of the [[braid group]]s, named after and originally studied by the German mathematician [[Werner Burau]]<ref name="burau">{{cite journal|last=Burau|first=Werner|year=1936|title=Über Zopfgruppen und gleichsinnig verdrillte Verkettungen|journal=Abh. Math. Sem. Hamburg|volume=11|pages=179−186}}</ref> during the 1930s. The '''Burau representation''' has two common and near-equivalent formulations, the '''reduced''' and '''unreduced''' Burau representations.
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== Definition ==
 
Consider the [[braid group]] <math>B_n</math> to be the [[mapping class group]] of a disc with ''n'' marked points <math>P_n</math>.  The [[homology group]] <math>H_1 P_n</math> is free abelian of rank ''n''.  Moreover, the invariant subspace of <math>H_1 P_n</math> (under the action of <math>B_n</math>) is primitive and infinite cyclic. Let <math>\pi : H_1 P_n \to \Bbb Z</math> be the projection onto this invariant subspace. Then there is a [[covering space]] <math>\tilde P_n</math> corresponding to this projection map. Much like in the construction of the [[Alexander polynomial]], consider <math>H_1 \tilde P_n</math> as a module over the group-ring of covering transformations <math>\Bbb Z[\Bbb Z] \equiv \Bbb Z[t^\pm]</math> (a [[Laurent polynomial|Laurent polynomial ring]]). As such a <math>\Bbb Z[t^\pm]</math>-module, <math>H_1 \tilde P_n</math> is free of rank ''n''&nbsp;&minus;&nbsp;1. By the basic theory of [[covering space]]s, <math>B_n</math> acts on <math>H_1 \tilde P_n</math>, and this representation is called the ''reduced Burau representation''.
 
The ''unreduced Burau representation'' has a similar definition, namely one replaces <math>P_n</math> with its [[blowing up|(real, oriented) blow-up]] at the marked points. Then instead of considering <math>H_1 \tilde P_n</math> one considers the relative homology <math>H_1 (\tilde P_n, \tilde \partial)</math> where <math>\partial \subset P_n</math> is the part of the boundary of <math>P_n</math> corresponding to the blow-up operation together with one point on the disc's boundary. <math>\tilde \partial</math> denotes the lift of <math>\partial</math> to <math>\tilde P_n</math>. As a <math>\Bbb Z[t^\pm]</math>-module this is free of rank ''n''. 
 
By the [[homology (mathematics)|homology long exact sequence of a pair]], the Burau representations fit into a short exact sequence <math>0 \to V_r \to V_u \to D \oplus \Bbb Z[t^\pm] \to 0</math>, where <math>V_r</math> and <math>V_u</math> are reduced and unreduced Burau <math>B_n</math>-modules respectively and <math>D \subset \Bbb Z^n</math> is the complement to the diagonal subspace (i.e.: <math>D = \{(x_1,\cdots,x_n) \in \Bbb Z^n : x_1+x_2+\cdots+x_n=0\}</math>, and <math>B_n</math> acts on <math>\Bbb Z^n</math> by the permutation representation.
 
== Relation to the Alexander polynomial ==
 
If a knot <math>K</math> is the closure of a braid <math>f</math>, then the [[Alexander polynomial]] is given by <math>\Delta_K(t) = \det(I-f_*)</math> where <math>f_*</math> is the reduced Burau representation of the braid <math>f</math>.
 
== Faithfulness ==
 
The first nonfaithful Burau representations are found without the use of computer, using a notion of winding number or contour integration.
<ref>[[J. Moody]], [http://www.jstor.org/pss/2159956 The faithfulness question for the Burau representation, Proc. AMS 1993]</ref> A more conceptual understanding <ref name="lp">
D D Long, M Paton, The Burau representation is not faithful for n &ge;  6, Topology 32 (1993)</ref> interprets the linking or winding as coming from [[Poincaré duality]] in first homology relative to the basepoint of a covering space, and uses the [[Poincaré duality|intersection form]] (traditionally called Squier's Form as Craig Squier was the first to explore its properties).<ref name="squier">{{cite journal|last=Squier|first=Craig C|year=1984|title=The Burau representation is unitary|journal=[[Proceedings of the American Mathematical Society]]|volume=90|issue=2|pages=199–202|doi=10.2307/2045338}}</ref>  Stephen Bigelow combined computer techniques and the Long-Paton theorem to show that the Burau representation is not faithful for n ≥ 5.<ref name="bigelow">{{cite journal|last=Bigelow|first=Stephen|year=1999|title=The Burau representation is not faithful for n = 5|journal=[[Geometry & Topology]]|volume=3|pages=397–404|doi=10.2140/gt.1999.3.397}}</ref>
<ref name="icm">[[S. Bigelow]],[http://www.ams.org/notices/200301/comm-icm2002.pdf International Congress of Mathematicians, Beijing, 2002]</ref><ref>[[V. Turaev]], [http://archive.numdam.org/article/SB_1999-2000__42__389_0.pdf Faithful representations of the braid groups, Bourbaki 1999-2000]</ref>
 
The Burau representation for ''n'' = 2,&nbsp;3 has been known to be faithful for some time. The faithfulness of the Burau representation when ''n'' = 4 is an open problem.
 
== Geometry ==
 
Squier showed that the Burau representation preserves a [[sesquilinear form]].<ref name="squier"/>  Moreover, when the variable <math>t</math> is chosen to be a transcendental unit [[complex number]] near <math>1</math> it is a positive-definite Hermitian pairing, thus the Burau representation can be thought of as a map into the [[Unitary group]].
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Burau Representation}}
[[Category:Braid groups]]
[[Category:Representation theory]]

Latest revision as of 18:16, 8 November 2014

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