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Scindements de Heegaard des espaces lenticulaires. (English) Zbl 0545.57002
A Heegaard splitting of a closed 3-manifold M is a surface F which separates M into two handlebodies. The authors show the following: Theorem 1. Up to isotopies, for each \(g\geq 1\) the lens space L(p,q) admits a unique Heegaard splitting of genus g. This gives an answer to Problem 3.22 in the paper by R. Kirby [Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)] given by Birman and Montesinos. F. Waldhausen [Topology 7, 195-203 (1968; Zbl 0157.545)] proved it for the case L(0,1), and F. Bonahon [Topology 22, 305-314 (1983; Zbl 0526.57009)] for the case \(g=1\). These two cases are used to prove the theorem. For this they examine the intersection of a Heegaard splitting F and the generalized projective plane \(\Delta\) in L(p,q) which is obtained as follows: The lens space L(p,q) is constructed by glueing together two copies \(V_ 1\) and \(V_ 2\) of \(S^ 1\times D^ 2\) via the map \(\theta\) : \(\partial V_ 1\to \partial V_ 2\) defined by \(\theta(u,v)=(u^ rv^ p,u^ sv^ q)\), where \(qr-ps=-1.\) Then \(\Delta\) is the union of the disk \(1\times D^ 2\) in \(V_ 1\) and the set \(\{(z^ p,\rho z^ q)|(z,\rho)\in S^ 1\times [0,1]\}\) in \(V_ 2\). They use the same technique as in W. Haken’s theorem [Stud. Math. 5 (Studies modern Topol.) 39-98 (1968; Zbl 0194.249)]: If G is a Heegaard splitting of a closed 3-manifold N containing an essential sphere, then there exists an essential sphere intersecting G by a loop. A Heegaard splitting F is said to be oriented, if a normal vector to F is settled. They also prove the following: Theorem 2. If \(g\geq 2\), or if \(g=1\) and q is congruent to \(\pm 1\) modulo p, then the lens space L(p,q) admits a unique oriented Heegaard splitting of genus g (up to isotopies). If q is not congruent to \(\pm 1\) modulo p, L(p,q) admits exactly two oriented Heegaard splittings of genus 1.
Reviewer: T.Nagase

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
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References:
[1] F. BONAHON , Difféotopies des espaces lenticulaires , à paraître dans Topology. Zbl 0526.57009 · Zbl 0526.57009 · doi:10.1016/0040-9383(83)90016-2
[2] F. BONAHON et J.-P. OTAL , Scindements de Heegaard des espaces lenticulaires (C.R. Acad. Sc., Paris, t. 294, série I, 1982 ). MR 83f:57008 | Zbl 0502.57006 · Zbl 0502.57006
[3] J. BIRMAN , F. GONZALES-ACUÑ;A , J.-M. MONTESINOS , Minimal Heegaard Splittings of 3-Manifolds Are Not Unique (Michigan Math. J., vol. 23, 1976 , p. 97-103). Article | Zbl 0321.57004 · Zbl 0321.57004 · doi:10.1307/mmj/1029001657 · minidml.mathdoc.fr
[4] R. ENGMANN , Nicht-homöomorphe Heegaard Zerlegungen von Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume , (Abh. Math. Sem. Hamburg, vol. 35, 1971 , p. 33-38). MR 44 #1033 | Zbl 0202.54601 · Zbl 0202.54601 · doi:10.1007/BF02992472
[5] W. HAKEN , Some Results on Surfaces in 3-Manifolds (Studies in Modern Topology, P. J. HILTON, éd., Prentice Hall, 1968 , p. 39-98). MR 36 #7118 | Zbl 0194.24902 · Zbl 0194.24902
[6] C. D. HODGSON , Involutions and Isotopies of Lens Spaces (Thèse de l’Université de Melbourne, 1981 ).
[7] R. KIRBY , Problems in Low-Dimensional Manifold Theory (Proceedings of Symposia in Pure Mathematics, vol. 32, Part 2, 1978 , 273-312). MR 80g:57002 | Zbl 0394.57002 · Zbl 0394.57002
[8] J.-P. OTAL , Présentations en ponts du nœud trivial (C.R. Acad. Sc., Paris, t. 294, série I, 1982 , p. 553-556). MR 84a:57006 | Zbl 0498.57001 · Zbl 0498.57001
[9] J.-P. OTAL , Scindements de Heegaard et présentations en ponts , (Thèse de 3e cycle, Orsay 1982 ).
[10] K. REIDEMEISTER , Zur dreidimensionalen Topologie (Abh. math. Semin. Univ. Hamburg, vol. 9, 1933 , p. 189-194). Zbl 0007.08005 | JFM 59.1240.01 · Zbl 0007.08005 · doi:10.1007/BF02940644 · www.emis.de
[11] K. REIDEMEISTER , Homotopieringe und Linsenräume (Abh. math. Semin. Univ. Hamburg, vol. 11, 1936 , p. 102-109). Zbl 0011.32404 | JFM 61.1352.01 · Zbl 0011.32404 · doi:10.1007/BF02940717 · www.emis.de
[12] H. SCHUBERT , Knoten mit zwei Brücken (Math. Zeit., vol. 65, 1956 , p. 133-170). MR 18,498e | Zbl 0071.39002 · Zbl 0071.39002 · doi:10.1007/BF01473875 · eudml:169591
[13] J. SINGER , Three-Dimensional Manifolds and Their Heegaard Diagrams , (Trans. A.M.S., vol. 35, 1933 , p. 88-111). MR 1501673 | Zbl 0006.18501 · Zbl 0006.18501 · doi:10.2307/1989314
[14] J. STALLINGS , On the Loop Theorem (Ann. of Math., vol. 72, 1960 , p. 12-19). MR 22 #12526 | Zbl 0094.36103 · Zbl 0094.36103 · doi:10.2307/1970146
[15] F. WALDHAUSEN , Heegaard-Zerlegungen der 3-Sphäre , (Topology, vol. 7, 1968 , p. 195-203). MR 37 #3576 | Zbl 0157.54501 · Zbl 0157.54501 · doi:10.1016/0040-9383(68)90027-X
[16] F. WALDHAUSEN , On Irreducible Manifolds which Are Sufficiently Large (Ann. of Math., vol. 87, 1968 , p. 56-88). MR 36 #7146 | Zbl 0157.30603 · Zbl 0157.30603 · doi:10.2307/1970594
[17] H. ZIESCHANG , Über einfache Kurven auf Vollbrezeln (Abh. math. Semin. Univ. Hamburg, vol. 25, 1962 ), p. 231-250). MR 26 #6957 | Zbl 0111.35801 · Zbl 0111.35801 · doi:10.1007/BF02992929
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