A strong Haken theorem

A T B is a Heegaard split compact orientable 3-manifold and S M is a reducing sphere for M .Haken (1968) showed that there is then also a reducing sphere S for the Heegaard splitting.Casson and Gordon (1987) extended the result to @-reducing disks in M and noted that in both cases S is obtained from S by a sequence of operations called 1-surgeries.Here we show that in fact one may take S D S . 57K35It is a foundational theorem of Haken [4 that any Heegaard splitting M D A T B of a closed orientable reducible 3-manifold M is reducible; that is, there is an essential sphere in the manifold that intersects T in a single circle.Casson and Gordon [1, Lemma 1.1 refined and generalized the theorem, showing that it applies also to essential disks, when M has boundary.More specifically, if S is a disjoint union of essential disks and 2-spheres in M then there is a similar family S , obtained from S by ambient 1-surgery and isotopy, such that each component of S intersects T in a single circle.In particular, if M is irreducible, so S consists entirely of disks, S is isotopic to S.There is of course a more natural statement, in which S does not have to be replaced by S .I became interested in whether the natural statement is true because it would be the first step in a program to characterize generators of the Goeritz group of S 3 ; see Freedman and the author 3;8.Inquiring of experts, I learned that this more natural statement had been pursued by some, but not successfully.Here we present such a proof.A reader who would like to get the main idea in a short amount of time could start with the example in Section 11.Recently, Hensel and Schultens 6 have proposed an alternative proof that applies when M is closed and S consists entirely of spheres.Here is an outline of the paper: Sections 1 and 2 are mostly a review of what is known; particularly the use of verticality in classical compression bodies, those which have no spheres in their boundary.We wish to allow sphere components in the boundary, and Section 3 explains how to recover the classical results in this context.Section 4 shows how to use these results to inductively reduce the proof of the main theorem to the case when S is connected.The proof when S is connected (the core of the proof) then occupies Sections 6 through 10.