Heegaard splittings of the form $H+nK$

Abstract

F. Waldhausen, in his 1978 paper [19], asked if every closed orientable three-manifold contains only finitely many unstabilized Heegaard splittings. A. Casson and C. Gordon (see [1] or [11]), using a result of R. Parris [13], obtain a definitive “no” answer; they obtain examples of closed hyperbolic three-manifolds each of which contains strongly irreducible splittings of arbitrarily large genus. These examples have been studied and generalized by T. Kobayashi [5], [6], M. Lustig and Y. Moriah [10], E. Sedgwick [17], and K. Hartshorn [3]. The goal of this paper is three-fold. We first show, in Section 3, that all of the examples studied so far are of the form H + nK: There is a pair of surfaces H and K in the manifold so that the strongly irreducible splittings are obtained via a cut-and-paste construction, Haken sum, of H with n copies of K. See Section 2 for a precise definition of Haken sum. Next, and of more interest, we show when such a sequence exists the surface K must be incompressible (in Sections 5 through 6). We claim: