Mapping class group of a handlebody

Bronisław Wajnryb

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 3, page 195-228
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Wajnryb, Bronisław. "Mapping class group of a handlebody." Fundamenta Mathematicae 158.3 (1998): 195-228. <http://eudml.org/doc/212312>.

@article{Wajnryb1998,
abstract = {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.},
author = {Wajnryb, Bronisław},
journal = {Fundamenta Mathematicae},
keywords = {handlebody; mapping class group; explicit presentation},
language = {eng},
number = {3},
pages = {195-228},
title = {Mapping class group of a handlebody},
url = {http://eudml.org/doc/212312},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Wajnryb, Bronisław
TI - Mapping class group of a handlebody
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 3
SP - 195
EP - 228
AB - 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.
LA - eng
KW - handlebody; mapping class group; explicit presentation
UR - http://eudml.org/doc/212312
ER -

References

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  1. [1] Bergau, P. und Mennicke, J., Über topologische Abbildungen der Bretzelfläche vom Geschlecht 2, Math. Z. 74 (1960), 414-435. 
  2. [2] Birman, J. S., Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton Univ. Press, 1974. 
  3. [3] Birman, J. S. and Hilden, H., On mapping class groups of closed surfaces as covering spaces, in: Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud. 66, Princeton Univ. Press, 1971, 81-115. Zbl0217.48602
  4. [4] Dehn, M., Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206. Zbl64.1276.01
  5. [5] Epstein, D. B. A., Curves on 2-manifolds and isotopies, ibid. 115 (1966), 83-107. Zbl0136.44605
  6. [6] Harer, J., The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), 221-239. Zbl0533.57003
  7. [7] Hatcher, A. and Thurston, W., A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221-237. Zbl0447.57005
  8. [8] Heusner, M., Eine Präsentation der Abbildungsklassengruppe einer geschlossenen, orientierbaren Fläche, Diplomarbeit, University of Frankfurt. 
  9. [9] Humphries, S., Generators for the mapping class group, in: Topology of Low-Dimensional Manifolds, Lecture Notes in Math. 722, Springer, 1979, 44-47. 
  10. [10] Johnson, D., Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), 119-125. Zbl0407.57003
  11. [11] Laudenbach, F., Présentation du groupe de difféotopies d'une surface compacte orientable, in: Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979), 267-282. 
  12. [12] Lickorish, W. B. R., A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769-778. Zbl0131.20801
  13. [13] Suzuki, S., On homeomorphisms of a 3-dimensional handlebody, Canad. J. Math. 29 (1977), 111-124. Zbl0339.57001
  14. [14] Wajnryb, B., A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), 157-174. Zbl0533.57002

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