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 pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of π which induce all allowable permutations of . 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 pointwise.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-1,
author = {Bronis\l aw Wajnryb and Agnieszka Wi\'sniowska-Wajnryb},
title = {Non-standard automorphisms of branched coverings of a disk and a sphere},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {1-11},
zbl = {1275.57003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-1}
}
Bronisław Wajnryb; Agnieszka Wiśniowska-Wajnryb. Non-standard automorphisms of branched coverings of a disk and a sphere. Fundamenta Mathematicae, Tome 219 (2012) pp. 1-11. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-1/