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Small 3-manifolds with large Heegaard distance

Published online by Cambridge University Press:  22 July 2013

TAO LI*
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467U.S.A. e-mail: taoli@bc.edu

Abstract

We construct examples of closed non-Haken hyperbolic 3-manifolds with a Heegaard splitting of arbitrarily large distance.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Agol, IanSmall 3-manifolds of large genus. Geom. Dedicata 102 (2003), 5364.CrossRefGoogle Scholar
[2]Finkelstein, E. and Moriah, Y.Closed incompressible surfaces in knot complements. Trans. Amer. Math. Soc. 352 (2000), 655677.CrossRefGoogle Scholar
[3]Floyd, W. and Hatcher, A.Incompressible surfaces in punctured-torus bundles. Topology Appl. 13 (1982), 263282.CrossRefGoogle Scholar
[4]Hartshorn, K.Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204 (2002), 6175.CrossRefGoogle Scholar
[5]Harvey, W. J.Boundary structure of the modular group. Ann. of Math. Stud. 97 (Princeton, 1981), 245251.Google Scholar
[6]Hatcher, A.On the boundary curves of incompressible surfaces. Pacific J. Math. 99 (1982), 373377.CrossRefGoogle Scholar
[7]Hatcher, A. and Thurston, W.Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79 (1985), 225249.CrossRefGoogle Scholar
[8]Hempel, J.3–manifolds as viewed from the curve complex. Topology 40 (2001), 631657.CrossRefGoogle Scholar
[9]Li, T.Heegaard surfaces and measured laminations, I: the Waldhausen conjecture. Invent. Math. 167 (2007), 135177.CrossRefGoogle Scholar
[10]Li, T.Heegaard surfaces and measured laminations, II: non-Haken 3–manifolds. J. Amer. Math. Soc. 19 (2006), 625657.CrossRefGoogle Scholar
[11]Li, T.Saddle tangencies and the distance of Heegaard splittings. Algebr. Geom. Topol. 7 (2007), 11191134.CrossRefGoogle Scholar
[12]Li, T.Heegaard surfaces and the distance of amalgamation. Geome. Topol. 14 (2010), 18711919.CrossRefGoogle Scholar
[13]Lustig, M. and Moriah, Y.Closed incompressible surfaces in complements of wide knots and links. Topology Appl. 92 (1999), 113.CrossRefGoogle Scholar
[14]Lustig, M. and Moriah, Y.High distance Heegaard splittings via fat train tracks. Topology Appl. 156 (2009), 11181129.CrossRefGoogle Scholar
[15]Masur, H. and Minsky, Y.Geometry of the complex of curves I: Hyperbolicity. Invent. Math. 138 (1999), 103149.CrossRefGoogle Scholar
[16]Oertel, U.Closed incompressible surfaces in complements of star links. Pacific J. Math. 111 (1984), 209230.CrossRefGoogle Scholar
[17]Scharlemann, M. and Tomova, M.Alternate Heegaard genus bounds distance. Geome. Topol. 10 (2006), 593617.CrossRefGoogle Scholar
[18]Waldhausen, F.On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 (1968), 5688.CrossRefGoogle Scholar