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AN ESTIMATION OF HEMPEL DISTANCE BY USING REEB GRAPH

AYAKO IDO

Department of Mathematics, Nara Women's University, Kitauoya Nishimachi, Nara 630-8506, Japan

https://doi.org/10.1142/S0218216511009327Cited by:1 (Source: Crossref)

Abstract

Let P, Q be Heegaard surfaces of a closed orientable 3-manifold. In this paper, we introduce a method for giving an upper bound of (Hempel) distance of P by using the Reeb graph derived from a certain horizontal arc in the ambient space [0, 1] × [0, 1] of the Rubinstein–Scharlemann graphic derived from P and Q. This is a refinement of a part of Johnson's arguments used for determining stable genera required for flipping high distance Heegaard splittings.

Keywords:

AMSC: 57M27, 57M50

References

M. Golubitsky and V. Guillemin , Stable Mappings and Their Singularities , Graduate Texts in Mathematics 14 ( Springer-Verlag , New York , 1973 ) . Crossref, Google Scholar
W. J. Harvey, Riemann Surfaces and Related Topics (Princeton University Press, Princeton, NJ, 1981) pp. 245–251. Crossref, Google Scholar
J. Hass, A. Thompson and W. Thurston, Stabilization of Heegaard splittings, preprint (2008) math.GT , arXiv:0802.2145v2 . Google Scholar
J. Hempel, Topology 40(3), 631 (2001), DOI: 10.1016/S0040-9383(00)00033-1. Crossref, Web of Science, Google Scholar
M. W. Hirsch , Differential Topology , Graduate Texts in Mathematics 33 ( Springer-Verlag , New York , 1976 ) . Crossref, Google Scholar
J. Johnson, Flipping and stabilizing Heegaard splittings, preprint (2008) math.GT , arXiv:0805.4422 . Google Scholar
J. Johnson, Trans. Amer. Math. Soc. 361(7), 3747 (2009), DOI: 10.1090/S0002-9947-09-04731-X. Crossref, Web of Science, Google Scholar
T. Kobayashi and O. Saeki, Pacific J. Math. 195(1), 101 (2000), DOI: 10.2140/pjm.2000.195.101. Crossref, Web of Science, Google Scholar
T. Li, Algebr. Geom. Topol. 7, 1119 (2007), DOI: 10.2140/agt.2007.7.1119. Crossref, Web of Science, Google Scholar
J. N. Mather, Adv. Math. 4, 301 (1970). Crossref, Web of Science, Google Scholar
H. Rubinstein and M. Scharlemann, Topology 35(4), 1005 (1996), DOI: 10.1016/0040-9383(95)00055-0. Crossref, Web of Science, Google Scholar
M. Scharlemann and M. Tomova, Geom. Topol. 10, 593 (2006), DOI: 10.2140/gt.2006.10.593. Crossref, Web of Science, Google Scholar
F. Waldhausen, Ann. Math. 87, 56 (1968), DOI: 10.2307/1970594. Crossref, Web of Science, Google Scholar