Lifting of homeomorphisms to branched coverings of a disk

Abstract

We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bn, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup L π of finite index in Bn. For each equivalence class of simple, d-sheeted coverings π of D with n branch values we find an explicit small set generating L π . The generators are powers of half-twists.

Non-standard automorphisms of branched coverings of a disk and a sphere

“…, B d } pointwise fixed. In [WW2] we have described a set of generators for the group L(π) of classes of homeomorphisms of D which lift to standard automorphisms of X.…”

“…Apostolakis considered 4-sheeted coverings in [A] and found generators for a certain quotient of the group L(π). In [WW1] a small finite set of generators of L(π) was found for every simple 4-sheeted covering of a disk and in [WW2] for simple coverings of any degree.…”