The word problem and consequences for the braid groups and mapping class groups of the 2-sphere

Abstract

to the presentation of Bn(E2), and showed that finite order braids exist for each n > 1. Magnus [10] derived a presentation of the group Mn of mapping classes of the n-punctured 2-sphere from that of Bn(S2) by adding the relation (u1l2 * * . an1)n = 1. A mapping class of the n-punctured 2-sphere can be visualized as an n-string braid between concentric 2-spheres, where the inner sphere is free to execute full revolutions. In this paper we modify Artin's effective procedure for putting a braid on E2 into normal form in order to solve the word problems for Bn(S2) and Mn. The resulting normal forms are then used to prove the following theorems.