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1

Mapping class group of a handlebody

100%

Wajnryb B.

Fundamenta Mathematicae

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1998

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tom 158

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nr 3

195-228

EN

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.

2

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

63%

Wajnryb B., Wiśniowska-Wajnryb A.

Fundamenta Mathematicae

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2012

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tom 218

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nr 1

1-11

EN

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 $π^{-1}(A₀)$ pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of π which induce all allowable permutations of $π^{-1}(A₀)$. 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 $π^{-1}(A₀)$ pointwise.

3

Lifting of homeomorphisms to branched coverings of a disk

63%

Wajnryb B., Wiśniowska-Wajnryb A.

Fundamenta Mathematicae

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2012

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tom 217

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nr 2

95-122

EN

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 of simple, d-sheeted coverings π of D with n branch values we find an explicit small set generating $L^π$. The generators are powers of half-twists.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Wajnryb B.

2 Wiśniowska-Wajnryb A.

2 2012

1 1998

Mapping class group of a handlebody

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

Lifting of homeomorphisms to branched coverings of a disk