Lifting of homeomorphisms to branched coverings of a disk

Bronisław Wajnryb ¹   ; Agnieszka Wiśniowska-Wajnryb ¹

¹ Department of Mathematics Rzeszów University of Technology Powstańców Warszawy 12 35-959 Rzeszów, Poland

Fundamenta Mathematicae, Tome 217 (2012) no. 2, pp. 95-122

Résumé

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.

Détail

DOI : 10.4064/fm217-2-1

Keywords: consider simple possibly disconnected d sheeted branched covering closed dimensional disk surface isotopy classes homeomorphisms which pointwise fixed boundary permute branch values form braid group where number branch values these homeomorphisms lifted homeomorphisms which fix pointwise fiber base point form subgroup finite index each equivalence class simple d sheeted coverings branch values explicit small set generating generators powers half twists