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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:
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