Non-standard automorphisms of branched coverings of a disk and a sphere
Volume 218 / 2012
Abstract
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.
