Genus 2 Heegaard decompositions of small Seifert manifolds
Annales de l'Institut Fourier, Volume 41 (1991) no. 4, p. 1005-1024
The genus 2 Heegaard splittings and decompositions of Seifert manifolds over S with 3 exeptional fibres are classified with respect to isotopies and homeomorphisms. In general there are 3 different isotopy classes of Heegaard splittings and 6 different isotopy classes of Heegaard decompositions. Moreover, we determine when a homeomorphism class is not an isotopy class.
Les scindements et décompositions de Heegaard de genre 2 des variétés de Seifert de base S ayant 3 fibres exceptionnelles sont classifiés à isotopie et homéomorphisme près. En général il y a 3 classes d’isotopie distinctes de scindements de Heegaard et 6 classes d’isotopie distinctes de décompositions de Heegaard. De plus on détermine précisément les classes d’homéomorphie qui ne sont pas des classes d’isotopie.
@article{AIF_1991__41_4_1005_0,
     author = {Boileau, Michel and Collins, D. J. and Zieschang, H.},
     title = {Genus 2 Heegaard decompositions of small Seifert manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {41},
     number = {4},
     year = {1991},
     pages = {1005-1024},
     doi = {10.5802/aif.1282},
     mrnumber = {93d:57026},
     zbl = {0734.57003},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1991__41_4_1005_0}
}
Genus 2 Heegaard decompositions of small Seifert manifolds. Annales de l'Institut Fourier, Volume 41 (1991) no. 4, pp. 1005-1024. doi : 10.5802/aif.1282. https://aif.centre-mersenne.org/item/AIF_1991__41_4_1005_0/

[1] J. S. Birman, F. Gonzáles-Acuña, J. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Mich. Math., J., 23 (1976), 97-103. | Zbl 0321.57004

[2] M. Boileau, D. J. Collins, H. Zieschang, Scindements de Heegaard des petites variétés de Seifert, C.R. Acad. Sci. Paris, 305-I (1987), 557-560. | MR 88k:57003 | Zbl 0651.57010

[3] M. Boileau, J.-P. Otal, Groupes des difféotopies de certaines variétés de Seifert. C.R. Acad. Sci., Paris, 303-I (1986), 19-22. | MR 87g:57022 | Zbl 0596.57010

Groupes d'homéotopies et scindements de Heegaard des petites variétés de Seifert, Invent. Math., 106 (1991), 85-107. | Zbl 0752.57006

[4] M. Boileau, J.-P. Otal, Sur les scindements de Heegaard du tore T3, J. Diff. Geom., 32 (1990), 209-233. | MR 91i:57006 | Zbl 0754.53012

[5] M. Boileau, M. Rost, H. Zieschang, On Heegaard decompositions of torus knot exteriors' and related Seifert fibre spaces, Math. Ann., 279 (1988), 553-581. | MR 89a:57013 | Zbl 0616.57008

[6] M. Boileau, H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math., 76 (1984), 455-468. | MR 86a:57008 | Zbl 0538.57004

[7] F. Bonahon, Difféotopies des espaces lenticulaires, Topology, 22 (1983), 305-314. | MR 85d:57008 | Zbl 0526.57009

[8] F. Bonahon, J.-P. Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Scient. Éc. Norm. Sup., 16 (1983), 451-466. | Numdam | MR 85c:57010 | Zbl 0545.57002

[9] M. Lustig, Nielsen equivalence and simple homotopy type, Proc. London Math. Soc. (to appear). | Zbl 0742.57003

[10] M. Lustig, Y. Moriah, Nielsen equivalence in Fuchsian groups and Seifert fibre spaces, Topology (to appear). | Zbl 0726.55010

[11] R. C. Lyndon, P. E. Schupp, Combinatorial Group Theory, Berlin-Heidelberg, New York, Springer, 1977. | MR 58 #28182 | Zbl 0368.20023

[12] Y. Moriah, Heegaard splittings and group presentations, Thesis, University of Texas, Austin, 1986.

[13] Y. Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math., 91 (1988), 465-481. | MR 89d:57010 | Zbl 0651.57012

[14] J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann., 78 (1918), 385-397. | JFM 46.0175.01

[15] P. Orlik, E. Vogt, H. Zieschang, Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten, Topology, 6 (1967), 49-64. | MR 35 #3696 | Zbl 0147.23503

[16] G. Rosenberger, All generating pairs of all two generator Fuchsian groups, Arch. Math., 46 (1986), 198-204. | MR 87g:20080 | Zbl 0563.20043

[17] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math., 60 (1933), 147-238. | JFM 59.1241.02 | Zbl 0006.08304

[18] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, Invent. Math., 3 (1967), 308-333 ; II, Invent. Math., 4 (1967), 87-117. | MR 38 #3880 | Zbl 0168.44503

[19] F. Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology, 7 (1968), 195-203. | MR 37 #3576 | Zbl 0157.54501

[20] H. Zieschang, B. Zimmermann, Über Erweiterungen von Z und Z * Z durch nichteuklidische kristallographische Gruppen, Math. Ann., 259 (1982), 29-51. | MR 84i:57001 | Zbl 0466.57002