Generators for two groups related to the braid group

Hugh M. Hilden

Vol. 59 (1975), No. 2, 475–486

DOI: 10.2140/pjm.1975.59.475

Abstract

Let D be the unit ball in 3-space and let A_k be a set of k proper disjoint arcs in D lying in the x − z plane. The group, ℳ_2k, of orientation preserving homeomorphisms of boundary D leaving the set A_k∩ boundary D invariant, modulo those isotopic to the identity via an isotopy fixing the set A_k∩ boundary D, is a natural homomorphic image of the 2k string braid group of the sphere via a homomorphism with kernel Z₂.

In this paper, finite sets of generators are explicitly determined for the subgroups of ℳ_2k generated by

(1) homeomorphisms of D leaving the set A_k invariant, and

(2) homeomorphisms of D leaving the set A_k fixed pointwise.

Mathematical Subject Classification

Primary: 55A25

Milestones

Received: 26 August 1974

Published: 1 August 1975