Non-standard automorphisms of branched coverings of a disk and a sphere (Q2895962)

For a simple branched covering \(\pi:X \rightarrow Y\) of a disk or sphere \(Y\), the authors study the automorphisms of \(\pi\), the homeomorphisms of \(Y\) which lift to \(X\). An automorphism induces a permutation of the points of the fiber over the base point of \(Y\). For the case \(Y\) a disk, the authors explicitly construct a set of automorphisms whose induced permutations generate all permutations which are realizable as an induced permutation. For the case \(Y\) a sphere, they show that every automorphism is isotopic to an automorphism whose induced permutation is trivial, by fiber-preserving isotopy.