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Mapping class group of a handlebody

Bronisław Wajnryb

1998
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Fundamenta Mathematicae
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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. [195] MCG(S) was done by Hatcher and Thurston in [7] where they constructed a connected and simply-connected cellular complex-a
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... ut-system complex of S-on which MCG(S) acts by cellular transformations with a finite number of orbits and well understood stabilizers. A presentation of MCG(S) was obtained from this action by methods described more explicitly by Laudenbach [11] . The presentation was simplified by Harer [6] , and further simplified by Wajnryb [14] . No presentation of M g has been known until now. A finite set of generators for M g was first obtained by Suzuki [13] . It was quite similar to the set of generators in our Theorem 18, the main result of this paper. The group M g has an infinite index in MCG(S) so the knowledge of MCG(S) does not help directly in the investigation of M g . However, we apply the main idea of Hatcher and Thurston and construct a simplicial, 2-dimensional cut-system complex X of B, similar to the cut-system complex of S, on which M g acts by simplicial transformations. We prove by a direct method, quite different from the method of Hatcher and Thurston, that X is connected and simply-connected. We describe the orbits of the action of M g on vertices, edges and faces of X. M g acts transitively on vertices and has only a finite number of edge-orbits and face-orbits. We describe the stabilizer of a vertex and the stabilizer of each edge and then apply the ideas from [7] and [11] to obtain an explicit presentation of M g (Theorem 18). We follow the Master dissertation of Michael Heusner [8] in which the method of [7] and [11] was very clearly and precisely explained. Unfortunately, an explicit presentation of M g (Theorem 18) obtained in this way is still rather long and complicated.

doi:10.4064/fm-158-3-195-228
fatcat:wdi47sybxfe7hg3qwd5sbncw5y