Two questions on Heegaard diagrams of $S^3$
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- by José María Montesinos
- Proc. Amer. Math. Soc. 102 (1988), 421-425
- DOI: https://doi.org/10.1090/S0002-9939-1988-0921010-6
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Abstract:
We review some of the methods that have been used to recognize ${S^3}$ from a Heegaard diagram. We propose a revision of these methods and examine their failure for manifolds different from ${S^3}$.References
- Joan S. Birman and José María Montesinos, On minimal Heegaard splittings, Michigan Math. J. 27 (1980), no. 1, 47–56. MR 555836
- Wolfgang Haken, Various aspects of the three-dimensional Poincaré problem, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 140–152. MR 0273624 —, Private conversation, 24-VII-80.
- Tatsuo Homma, Mitsuyuki Ochiai, and Moto-o Takahashi, An algorithm for recognizing $S^{3}$ in $3$-manifolds with Heegaard splittings of genus two, Osaka Math. J. 17 (1980), no. 3, 625–648. MR 591141
- Takeshi Kaneto, A note on an example of Birman-Montesinos, Proc. Amer. Math. Soc. 83 (1981), no. 2, 425–426. MR 624945, DOI 10.1090/S0002-9939-1981-0624945-0
- Takeshi Kaneto, On genus $2$ Heegaard diagrams for the $3$-sphere, Trans. Amer. Math. Soc. 276 (1983), no. 2, 583–597. MR 688963, DOI 10.1090/S0002-9947-1983-0688963-5
- Osamu Morikawa, A counterexample to a conjecture of Whitehead, Math. Sem. Notes Kobe Univ. 8 (1980), no. 2, 295–298. MR 601897
- 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
- O. Ja. Viro and V. L. Kobel′skiĭ, The Volodin-Kuznecov-Fomenko conjecture on Heegaard diagrams is false, Uspehi Mat. Nauk 32 (1977), no. 5(197), 175–176 (Russian). MR 0467757 F. Waldhausen, Some problems on $3$-manifolds, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1977.
- Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X
- J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1936), no. 4, 782–800. MR 1503309, DOI 10.2307/1968618 —, On certain sets of elements in a free group, London Math. Soc. 41 (1936), 48-56. I. A. Volodin, V. E. Kuznetsov and A. T. Fomenko, The problem of discriminating algorithmically the standard three-dimensional sphere, Russian Math. Survey 29 (1974), 71-172. H. Zieschang, On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. 92 (1970), 127-137.
- R. P. Osborne, Heegaard diagrams of lens spaces, Proc. Amer. Math. Soc. 84 (1982), no. 3, 412–414. MR 640243, DOI 10.1090/S0002-9939-1982-0640243-4
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 421-425
- MSC: Primary 57M40,; Secondary 57N12
- DOI: https://doi.org/10.1090/S0002-9939-1988-0921010-6
- MathSciNet review: 921010