Heegaard splittings and singularities of the product map of Morse functions

Takao, Kazuto

doi:10.1090/S0002-9947-2013-06015-1

Heegaard splittings and singularities of the product map of Morse functions
HTML articles powered by AMS MathViewer

by Kazuto Takao PDF

Trans. Amer. Math. Soc. 366 (2014), 2209-2226 Request permission

Abstract:

We give an upper bound for the Reidemeister–Singer distance between two Heegaard splittings in terms of the genera and the number of cusp points of the product map of Morse functions for the splittings. It suggests that a certain development in singularity theory may lead to the best possible bound for the Reidemeister–Singer distance.

References

D. Bachman: Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization, arXiv:math/0903.1695.
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
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
Joel Hass, Abigail Thompson, and William Thurston, Stabilization of Heegaard splittings, Geom. Topol. 13 (2009), no. 4, 2029–2050. MR 2507114, DOI 10.2140/gt.2009.13.2029
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Jesse Johnson, Stable functions and common stabilizations of Heegaard splittings, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3747–3765. MR 2491898, DOI 10.1090/S0002-9947-09-04731-X
Jesse Johnson, Bounding the stable genera of Heegaard splittings from below, J. Topol. 3 (2010), no. 3, 668–690. MR 2684516, DOI 10.1112/jtopol/jtq021
J. Johnson: An upper bound on common stabilizations of Heegaard splittings, arXiv:math/1107.2127.
Tsuyoshi Kobayashi and Osamu Saeki, The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map, Pacific J. Math. 195 (2000), no. 1, 101–156. MR 1781617, DOI 10.2140/pjm.2000.195.101
Harold I. Levine, Elimination of cusps, Topology 3 (1965), no. suppl, suppl. 2, 263–296. MR 176484, DOI 10.1016/0040-9383(65)90078-9
John N. Mather, Stability of $C^{\infty }$ mappings. V. Transversality, Advances in Math. 4 (1970), 301–336 (1970). MR 275461, DOI 10.1016/0001-8708(70)90028-9
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
K. Reidemeister: Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 11 (1933), 189-194.
Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings of non-Haken $3$-manifolds, Topology 35 (1996), no. 4, 1005–1026. MR 1404921, DOI 10.1016/0040-9383(95)00055-0
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
James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. MR 1501673, DOI 10.1090/S0002-9947-1933-1501673-5
Kazuto Takao, A refinement of Johnson’s bounding for the stable genera of Heegaard splittings, Osaka J. Math. 48 (2011), no. 1, 251–268. MR 2802601