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Mapping class group of a handlebody

Bronisław Wajnryb — 1998

Fundamenta Mathematicae

Let B be a 3-dimensional handlebody of genus g. Let ℳ be the group of the isotopy classes of orientation preserving homeomorphisms of B. We construct a 2-dimensional simplicial complex X, connected and simply-connected, on which ℳ acts by simplicial transformations and has only a finite number of orbits. From this action we derive an explicit finite presentation of ℳ.

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

Bronisław WajnrybAgnieszka Wiśniowska-Wajnryb — 2012

Fundamenta Mathematicae

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 - 1 ( A ) pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of...

Lifting of homeomorphisms to branched coverings of a disk

Bronisław WajnrybAgnieszka Wiśniowska-Wajnryb — 2012

Fundamenta Mathematicae

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 L π of finite index in Bₙ. For each equivalence class...

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