No 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.

No abstract

Let p : X → D be a simple, possibly not connected, 4-sheeted branched covering of a closed 2-dimensional disk D with n branch values A 1 , . . . , A n . The isotopy classes of homeomorphisms of D which are fixed on the boundary of D and permute the branch values form a braid group B n . Some of these homeomorphisms can be lifted to homeomorphisms of X. They form a subgroup L(p) of finite index in B n . For each equivalence class of coverings we find a set of generators for L(p) which contains between n and n + 4 elements, depending on the equivalence class of the covering, and the generators are powers of half-twists.

Let Y be a closed 2-dimensional disk or a 2-sphere. We consider a simple, d-sheeted branched covering π : X → Y . We fix a base point A0 in Y (A0 ∈ ∂Y if Y is a disk). We consider the homeomorphisms h of Y which fix ∂Y pointwise and lift to homeomorphisms φ of X-the automorphisms of π. We prove that if Y is a sphere then every such φ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber π −1 (A0) pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of π which induce all allowable permutations of π −1 (A0). This complements our result in Fund. Math. 217 (2012), no. 2, where we found a set of generators for the group of isotopy classes of automorphisms of π which fix the fiber π −1 (A0) pointwise.