SCIENCE CHINA Mathematics October 2020 Vol. 63 No. 10: 1997–2004 https://doi.org/10.1007/s11425-019-1647-9 c ⃝ Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 math.scichina.com link.springer.com Progress of Projects Supported by NSFC . ARTICLES . A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody Fengchun Lei 1 , He Liu 1 , Fengling Li 1,* & Andrei Vesnin 2 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China; 2 Regional Scientific and Educational Mathematical Center, Department of Mechanics and Mathematics, Tomsk State University, Tomsk 634050, Russia Email: fclei@dlut.edu.cn, 1033410708@qq.com, fenglingli@dlut.edu.cn, vesnin@math.nsc.ru Received August 27, 2019; accepted January 23, 2020; published online March 20, 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 = H 1 ∪ A H 2 of two handlebodies H 1 and H 2 is a handlebody if and only if the core curve of A is a longitude for either H 1 or H 2 ; (2) Let H = H 1 ∪ S g,b H 2 be a surface sum of two handlebodies H 1 and H 2 along a connected surface S = S g,b , b  1, n i = g(H i )  2, i =1, 2. Suppose that S is incompressible in both H 1 and H 2 . Then H is a handlebody if and only if there exists a basis J = {J 1 ,...,Jm} with a partition (J 1 , J 2 ) of J such that J 1 is primitive in H 1 and J 2 is primitive in H 2 . Keywords handlebody, surface sum of 3-manifolds, free group MSC(2010) 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: 1997–2004, https://doi.org/10.1007/s11425-019-1647-9 1 Introduction Let M 1 and M 2 be two compact connected orientable 3-manifolds, F i ⊂ ∂M i be a compact connected surface, i =1, 2, and h : F 1 → F 2 be a homeomorphism. We call the 3-manifold M = M 1 ∪ h M 2 , obtained by gluing M 1 and M 2 together via h,a surface sum of M 1 and M 2 . M = M 1 ∪ F M 2 also means a surface sum of M 1 and M 2 along F , where F = F 1 = F 2 . When F i is a boundary component of M i , i =1, 2, M is called an amalgamated 3-manifold of M 1 and M 2 along F = F 1 = F 2 . 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 quasi- Fuchsian 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