Faithfulness Properties of the Burau Representation

Caleb Hussey

Date of Award

2005

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Mullins, David

Keywords

Burau Representation, Braid Group/Faithful

Area of Concentration

Mathematics

Abstract

This paper is intended as a unified treatment of the existing results on the faithfulness of the Burau representation. Such a treatment is desirable because the articles cited often use conflicting conventions, sometimes even within a single paper. We present the results in terms of a single set of conventions, and we clarify or clean up proofs as needed. We introduce the n-strand braid group Bn as a set of homotopy classes of maps on Dn. We then define a covering space Dn of Dn and note that as we defined it, Bn acts naturally on the relative homology of Dn . We use this action to construct the Burau representation of Bn. Switching to a reduced form of the representation allows us to prove the Burau representation is faithful for n < 3. The representation is shown to be unfaithful for n > 5 by means of a curve in D5 with certain homological properties.

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