A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody
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https://doi.org/10.1007/s11425-019-1647-9
A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody Fengchun Lei1 , He Liu1 , Fengling Li1,∗ & Andrei Vesnin2 1School 2Regional
of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China; Scientific and Educational Mathematical Center, Department of Mechanics and Mathematics, Tomsk State University, Tomsk 634050, Russia
Email: [email protected], [email protected], [email protected], [email protected] Received August 27, 2019; accepted January 23, 2020
Abstract
The main results of the paper are that we give a necessary and sufficient condition for a surface sum
of two handlebodies along a connected surface to be a handlebody as follows: (1) The annulus sum H = H1 ∪A H2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2 ; (2) Let H = H1 ∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S = Sg,b , b > 1, ni = g(Hi ) > 2, i = 1, 2. Suppose that S is incompressible in both H1 and H2 . Then H is a handlebody if and only if there exists a basis J = {J1 , . . . , Jm } with a partition (J1 , J2 ) of J such that J1 is primitive in H1 and J2 is primitive in H2 . Keywords MSC(2010)
handlebody, surface sum of 3-manifolds, free group 57N10
Citation: Lei F C, Liu H, Li F L, et al. A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1647-9
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Introduction
Let M1 and M2 be two compact connected orientable 3-manifolds, Fi ⊂ ∂Mi be a compact connected surface, i = 1, 2, and h : F1 → F2 be a homeomorphism. We call the 3-manifold M = M1 ∪h M2 , obtained by gluing M1 and M2 together via h, a surface sum of M1 and M2 . M = M1 ∪F M2 also means a surface sum of M1 and M2 along F , where F = F1 = F2 . When Fi is a boundary component of Mi , i = 1, 2, M is called an amalgamated 3-manifold of M1 and M2 along F = F1 = F2 . Heegaard distances and related topics of amalgamation of two Heegaard splittings have been studied extensively in recent years (see, for example, [1, 6, 7, 9, 17, 18]). Composite knot complements are another important examples which are annulus sums of knot complements. In [9], some facts on Heegaard splittings of an annulus sum of 3-manifolds have been given, which played an essential role in calculating the Heegaard genus of the corresponding 3-manifold. Hyperbolic geometric structures related to quasiFuchsian realizations of once-punctured torus groups were studied in [12]. In [8], some properties of an annulus sum of 3-manifolds were obtained. In particular, a sufficient condition for an annulus sum of two handlebodies to be a handlebody was given. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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We use Sg,b to denote the connected compact orientable surface of ge
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