Lifting of homeomorphisms to branched coverings of a disk
Volume 217 / 2012
Fundamenta Mathematicae 217 (2012), 95-122
MSC: Primary 20F36, 20F38, 57M12.
DOI: 10.4064/fm217-2-1
Abstract
We consider a simple, possibly disconnected, $d$-sheeted branched covering $\pi$ 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 ${\bf B}_n$, 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 $L^\pi$ of finite index in ${\bf B}_n$. For each equivalence class of simple, $d$-sheeted coverings $\pi$ of $D$ with $n$ branch values we find an explicit small set generating $L^\pi$. The generators are powers of half-twists.
