3-fold branched coverings and the mapping class group of a surface. (English) Zbl 0589.57009

Birman, Joan S.; Wajnryb, Bronislaw

Geometry and topology, Proc. Spec. Year, College Park/Md. 1983-84, Lect. Notes Math. 1167, 24-46 (1985).

[For the entire collection see Zbl 0568.00014.]
Author’s summary: ”Let \(\rho\) : \(F\to D\) be a simple 3-sheeted branched covering of a 2-disc D, with an even number of branch values. Let L be the group of isotopy classes of liftable orientation-preserving homeomorphisms of D rel \(\partial D\). Then lifting induces a homeomorphism \(\lambda\) from L to the mapping class group of F. In this paper we prove that \(\lambda\) is surjective, and find a simple set of generators of L and two elements of L whose normal closure in L is kernel \(\lambda\). Thus the mapping class group of F is exhibited as a quotient group of the group L, which is a subgroup of finite index in Artin’s braid group”.

Reviewer: B.Zimmermann

MSC:

57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M12	Low-dimensional topology of special (e.g., branched) coverings
20F36	Braid groups; Artin groups

Keywords:

simple 3-sheeted branched covering; isotopy classes of liftable orientation-preserving homeomorphisms; mapping class group; Artin’s braid group

Citations:

Zbl 0568.00014