|
Stable functions and common stabilizations of Heegaard splittings
Author(s):
Jesse
Johnson
Journal:
Trans. Amer. Math. Soc.
361
(2009),
3747-3765.
MSC (2000):
Primary 57Mxx
Posted:
March 4, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type two cusps in a Rubinstein-Scharlemann graphic for the two splittings.
References:
-
- 1.
- David Bachman and Saul Schleimer, Surface bundles versus Heegaard splittings, preprint (2002). MR 2216145 (2006m:57027)
- 2.
- Jean Cerf, La stratefacation naturelle des especes de fonctions differentiables reeles et la theoreme de la isotopie, Publ. Math. I.H.E.S. 39 (1970).
- 3.
- Francesco Costantino and Dylan P. Thurston, 3-manifolds efficiently bound 4-manifolds, preprint (2005).
- 4.
- H. Edelsbrunner, Jacobi sets of multiple Morse functions, Foundations of computational mathematics: Minneapolis, 2002, Cambridge University Press, 2004, pp. 35-57. MR 2189626 (2006j:58014)
- 5.
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, 1973. MR 0341518 (49:6269)
- 6.
- A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221-237. MR 579573 (81k:57008)
- 7.
- Jesse Johnson, Heegaard splittings and the pants complex, Algebr. Geom. Topol. 6 (2006), 853-874. MR 2240918 (2007k:57039)
- 8.
- Tsuyoshi Kobayashi and Osamu Saeki, The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map, Pacific Journal of Mathematics 195 (2000), no. 1, 101-156. MR 1781617 (2001i:57026)
- 9.
- H. Levine, Classifying immersions into
 over stable maps of 3-manifolds into  , Springer-Verlag, 1980. MR 814689 (88f:57056) - 10.
- J. Mather, Stability of
 mappings V, Advances in Mathematics 4 (1970), no. 3, 301-336. MR 0275461 (43:1215c) - 11.
- J. Milnor, Morse theory, Annals of Mathematics Studies, no. 51, Princeton University Press, 1963. MR 0163331 (29:634)
- 12.
- A. Du Plessis and T. Wall, The geometry or topological stability, Oxford Science Publications, 1995. MR 1408432 (97k:58024)
- 13.
- K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 11 (1933), 189-194.
- 14.
- Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (1996), no. 4, 1005-1026. MR 1404921 (97j:57021)
- 15.
- -, Transverse Heegaard splittings, Michigan Math. J. 44 (1997), no. 1, 69-83. MR 1439669 (98c:57017)
- 16.
- Jennifer Schultens, Heegaard splittings of Seifert fibered spaces with boundary, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2533-2552. MR 1297537 (95j:57019)
- 17.
- J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111. MR 1501673
- 18.
- Friedhelm Waldhausen, Heegaard-Zerlegungen der
 -Sphäre, Topology 7 (1968), 195-203. MR 0227992 (37:3576)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
57Mxx
Retrieve articles in all Journals with MSC
(2000):
57Mxx
Additional Information:
Jesse
Johnson
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520
Email:
jessee.johnson@yale.edu
DOI:
10.1090/S0002-9947-09-04731-X
PII:
S 0002-9947(09)04731-X
Keywords:
Heegaard splitting,
stabilization,
Rubinstein-Scharlemann graphic
Received by editor(s):
May 30, 2007
Posted:
March 4, 2009
Additional Notes:
This research was supported by NSF MSPRF grant 0602368
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org