Lifting Braids

Abstract

In this paper we study the homeomorphisms of B$^{2}$ that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fi{}xing the branch set, we are dealing in fact with liftable braids. We prove that the group of liftable braids is fi{}nitely generated by liftable powers of half-twists around arcs joining branch points. A set of such generators is explicitly determined for the special case of branched coverings B$^{2}$ $\rightarrow$ B$^{2}$ . As a preliminary result we also obtain the classifi{}cation of all the simple branched coverings of B$^{2}$.