Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Comparing Heegaard splittings -the bounded case
HTML articles powered by AMS MathViewer

by Hyam Rubinstein and Martin Scharlemann PDF
Trans. Amer. Math. Soc. 350 (1998), 689-715 Request permission


In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If $p \leq q$ are the genera of two splittings, then there is a common stabilization of genus $5p + 8q - 9$. Here we show how to obtain similar results even when the 3-manifold has boundary.
  • 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
  • Jean Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$, Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin-New York, 1968 (French). MR 0229250, DOI 10.1007/BFb0060395
  • Charles Frohman, Minimal surfaces and Heegaard splittings of the three-torus, Pacific J. Math. 124 (1986), no. 1, 119–130. MR 850670, DOI 10.2140/pjm.1986.124.119
  • Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
  • K. Johannson, Topology and Combinatorics of 3-Manifolds, Lecture Notes in Math, 1599 Springer-Verlag, Berlin and New York, 1995.
  • H. Rubinstein and M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (1996), 1005–1026.
  • H. Rubinstein and M. Scharlemann, Transverse Heegaard splittings, Michigan Math. J. 49 (1997), 69–83.
  • J. Schultens, The stabilization problem for Heegaard splittings of Seifert fibered spaces, Topology and its Applications 73 (1996), 133-139.
  • 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
  • Martin Scharlemann and Abigail Thompson, Thin position for $3$-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231–238. MR 1282766, DOI 10.1090/conm/164/01596
  • Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 57N10, 57M50
  • Retrieve articles in all journals with MSC (1991): 57N10, 57M50
Additional Information
  • Hyam Rubinstein
  • Affiliation: Department of Mathematics, University of Melbourne, Parkville, Vic 3052, Australia
  • MR Author ID: 151465
  • Email:
  • Martin Scharlemann
  • Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
  • MR Author ID: 155620
  • Email:
  • Received by editor(s): December 21, 1995
  • Received by editor(s) in revised form: May 8, 1996
  • Additional Notes: Each author was partially supported by a grant from the Australian Research Council
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 689-715
  • MSC (1991): Primary 57N10; Secondary 57M50
  • DOI:
  • MathSciNet review: 1401528