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

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 218 (2012) no. 1, pp. 1-11

Résumé

Let $Y$ be a closed 2-dimensional disk or a 2-sphere. We consider a simple, $d$-sheeted branched covering $\pi :X\to Y$. We fix a base point $A_0$ in $Y$ ($A_0\in \partial Y$ if $Y$ is a disk). We consider the homeomorphisms $h$ of $Y$ which fix $\partial Y$ pointwise and lift to homeomorphisms $\phi $ of $X$—the automorphisms of $\pi $. We prove that if $Y$ is a sphere then every such $\phi $ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber $\pi ^{-1}(A_0)$ pointwise. If $Y$ is a disk, we describe explicitly a small set of automorphisms of $\pi $ which induce all allowable permutations of $\pi ^{-1}(A_0)$. 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 $\pi $ which fix the fiber $\pi ^{-1}(A_0)$ pointwise.

Détail

DOI : 10.4064/fm218-1-1

Keywords: closed dimensional disk sphere consider simple d sheeted branched covering fix base point partial disk consider homeomorphisms which fix partial pointwise lift homeomorphisms phi automorphisms prove sphere every phi isotopic fiber preserving isotopy automorphism which fixes fiber pointwise disk describe explicitly small set automorphisms which induce allowable permutations complements result fund math nbsp where found set generators group isotopy classes automorphisms which fix fiber pointwise