### Braid Groups and Their Relationship to Mapping Class Groups

New York University, 1968

Abstract

In 1962 R. H. Fox introduced a new definition of the braid group as the fundamental group of the space of n unordered distinct points of the Euclidean plane. Fox's definition suggested a natural generalization to the concept of a braid group on an arbitrary manifold, as the fundamental group of the space of n unordered, distinct points of that manifold. The present investigation begins with Fox's definition, and studies the algebraic and geometric properties of these braid groups on arbitrary manifolds.

In Part I a new meaning is given to the Fox braid groups, by relating them to the mapping class groups of the manifold.

Part 2 contains an algebraic investigation of the braid groups.

In Part 3 the algebraic connection between braid groups and mapping class groups is studied.