We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bₙ, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup of finite index in Bₙ. For each equivalence class of simple, d-sheeted coverings π of D with n branch values we find an explicit small set generating . The generators are powers of half-twists.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-2-1,
author = {Bronis\l aw Wajnryb and Agnieszka Wi\'sniowska-Wajnryb},
title = {Lifting of homeomorphisms to branched coverings of a disk},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {95-122},
zbl = {1254.57004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-2-1}
}
Bronisław Wajnryb; Agnieszka Wiśniowska-Wajnryb. Lifting of homeomorphisms to branched coverings of a disk. Fundamenta Mathematicae, Tome 219 (2012) pp. 95-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-2-1/