Handle Additions On 3-manifolds

Dehn surgery and Heegaard splitting are two important methods of constructing 3-manifolds. These two methods can be viewed as handle additions. As an important problemof handle additions, it is interesting to consider that how many 2-handle additions on a genusat least two boundary component of a hyperbolic 3-manifold can obtain a non-hyperbolicmanifold. Let M be a hyperbolic manifold, and F be a component of (?)M of genus at leasttwo. Supposeβandβare two separating slops on F. M. Scharlemann and Y. Wu provedthat if M(α) and M(β) are not hyperbolic, then the intersection numberΔ(α,β)≤14.Furthermore, they gave a conjecture: if M(α) and M(β) are reducible or (?)-reducible, thenΔ(α,β)=0. In this paper, we prove that if both M(α) and M(β) are reducible, thenΔ(α,β)≤2. Specially, if both M(α) and M(β) are reducible and the genus of F is 2, thenΔ(α,β)=0. This result together with the result in [35] and the result in [56] indicates thatthere is only one separating slopαon any component F of (?)M of genus two, such thatM(α) is either reducible or (?)-reducible. This means that the above conjecture is true whenthe genus of F is two. As a corollary of the above results, we also give a partial proof of theRefilling conjecture which is given by M. Scharlemann.Suppose M_i= V_i∪W_i (i =1, 2) are Heegaard splittings. A homeomorphism f: F₁→F₂produces an attached manifold M= M₁∪_F1=F2M₂, where F_i(?)_Wi. In this paper wedefine a surface sum of Heegaard splittings induced from the Heegaard splittings of M₁ andM₂. We also give an example showing that the surface sum of Heegaard splittings is differentfrom amalgamation of Heegaard splittings. Furthermore, we give a sufficient condition whenthe surface sum of Heegaard splittings is stabilized.Thin position of Heegaard splittings defined by M. Scharlemann and A. Thompsonplays an important role in the study of Heegaard splittings. Using Kneser-Haken's Theorem,we prove that any non-excessive set of completely disjoint bicompressible surfaces containsonly finitely elements. Furthermore, we prove that the length of any unstabilized Heegaardsplittings of a 3-manifold has an upper bound.