In this paper we study the homeomorphisms of the disk 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 fixing the branch set, we are dealing in fact with liftable braids. We
prove that the group of liftable braids is finitely 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
from the disk to the disk. As a preliminary result we also obtain the
classification of all the simple branched coverings of the disk.Comment: 20 page
