Scharlemann-Thompson untelescoping of Heegaard splittings is finer than Casson-Gordon’s

Let H1∪P H2 be a Heegaard splitting of a closed 3-manifoldM , i.e., Hi (i = 1, 2) is a handlebody in M such that H1 ∪H2 = M , H1 ∩H2 = ∂H1 = ∂H2 = P . In [13], M.Scharlemann, and A.Thompson had introduced a process for spreading H1 ∪P H2 into a “thinner” presentation. The idea was polished to show that if the original Heegaard splitting is irreducible, then we can spread it into a series (A1 ∪P1 B1) ∪ · · · ∪ (An ∪Pn Bn) such that each Ai ∪Pi Bi is a strongly irreducible Heegaard splitting. In this paper, we call this series of strongly irreducible Heegaard splittings a Scharlemann-Thompson untelescoping (or S-T untelescoping) of H1 ∪P H2. On the other hand, preceding [13], A.Casson, and C.Gordon [2] had proved that if H1∪P H2 is weakly reducible and not reducible, then there exists an incompressible surface of positive genus in M . This result is proved by using the following argument. Let ∆ = ∆1 ∪∆2 be a weakly reducing collection of compressing disks for P (for the definitions of the terms, see section 2). Then P (∆) denotes the surface obtained from P by compressing along ∆. Let P̂ (∆) be the surface obtained from P (∆) by discarding the components that are contained in either H1 or H2. Suppose that ∆ has minimal complexity (for the definition of the complexity, see section 4). Then we can show that the irreducibility of H1 ∪P H2 implies that no component of P̂ (∆) is a 2-sphere. Then, by using a relative version of Haken’s theorem [3], we can show that P̂ (∆) is incompressible. With adopting the above notations, we will see, in section 4, that the closure of each component of M − P̂ (∆) naturally inherits a Heegaard splitting from H1 ∪P H2 if P̂ (∆) contains no 2-sphere component. Hence we obtain a series of Heegaard splitings, say (C1 ∪Q1 D1) ∪ · · · ∪ (Cm ∪Qm Dm). If P̂ (∆) is incompressible, then this series is called a Casson-Gordon untelescoping (or C-G untelescoping) of H1 ∪P H2. Then we will also see that C-G untelescoping of H1 ∪P H2 can be regarded as one that appears in a process for obtaining S-T untelescoping from H1 ∪P H2 (Remark 4.1). We note that these two untelescopings are used in many articles and equally usefull. For example, C-G untelescoping was used by M.Boileau, and J.-P.Otal [1] for studying Heegaard splittings of the 3-dimensional torus, by J.Schultens [15] for studying Heegaard splittings of (surface)×S, by M.Lustig, and Y.Moriah [8] for studying the exteriors of wide knots and links, and by the author [7] for studying the Heegaard splittings of the exteriors of two bridge knots. S-T untelescoping was used, for example, by Scharlemann-Schultens [12], Schultens[14], Morimoto[9], and Morimoto-Schultens [10]for studying the Heegaard splittings