Heegaard splittings of Seifert fibered spaces with boundary

Schultens, Jennifer

doi:10.1090/S0002-9947-1995-1297537-5

Heegaard splittings of Seifert fibered spaces with boundary
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Trans. Amer. Math. Soc. 347 (1995), 2533-2552 Request permission

Abstract:

We give the classification theorem for Heegaard splittings of fiberwise orientable Seifert fibered spaces with nonempty boundary. A thin position argument yields a reducibility result which, by induction, shows that all Heegaard splittings of such manifolds are vertical in the sense of Lustig-Moriah. Algebraic arguments allow a classification of the vertical Heegaard splittings.

References

Genus

Heegaard decompositions of small Seifert manifolds

Michel Boileau and Jean-Pierre Otal, Groupe des difféotopies de certaines variétés de Seifert, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 1, 19–22 (French, with English summary). MR 849619
A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283. MR 918537, DOI 10.1016/0166-8641(87)90092-7
David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
Martin Lustig, Nielsen equivalence and simple-homotopy type, Proc. London Math. Soc. (3) 62 (1991), no. 3, 537–562. MR 1095232, DOI 10.1112/plms/s3-62.3.537
Martin Lustig and Yoav Moriah, Nielsen equivalence in Fuchsian groups and Seifert fibered spaces, Topology 30 (1991), no. 2, 191–204. MR 1098913, DOI 10.1016/0040-9383(91)90005-O
Yoav Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988), no. 3, 465–481. MR 928492, DOI 10.1007/BF01388781
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055
Martin Scharlemann and Abigail Thompson, Heegaard splittings of $(\textrm {surface})\times I$ are standard, Math. Ann. 295 (1993), no. 3, 549–564. MR 1204837, DOI 10.1007/BF01444902
Jennifer Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times \, S^1$, Proc. London Math. Soc. (3) 67 (1993), no. 2, 425–448. MR 1226608, DOI 10.1112/plms/s3-67.2.425
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, DOI 10.1007/BF02398271
Heiner Zieschang, Elmar Vogt, and Hans-Dieter Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, vol. 835, Springer, Berlin, 1980. Translated from the German by John Stillwell. MR 606743, DOI 10.1007/BFb0089692