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Let Y be a closed 2-dimensional disk or a 2-sphere. We consider a simple, d-sheeted branched covering π : X → Y . We fix a base point A0 in Y (A0 ∈ ∂Y if Y is a disk). We consider the homeomorphisms h of Y which fix ∂Y pointwise and lift to homeomorphisms φ of X—the automorphisms of π. We prove that if Y is a sphere then every such φ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber π(A0) pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of π which induce all allowable permutations of π(A0). 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 π which fix the fiber π(A0) pointwise.

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