ar X iv : m at h / 97 06 22 2 v 1 [ m at h . G T ] 3 0 Ju n 19 97 EXTENSION OF INCOMPRESSIBLE SURFACES ON THE BOUNDARY OF 3 - MANIFOLDS

An incompressible surface F on the boundary of a compact orientable 3manifold M is arc-extendible if there is an arc γ on ∂M − IntF such that F ∪ N(γ) is incompressible, where N(γ) is a regular neighborhood of γ in ∂M . Suppose for simplicity that M is irreducible, and F has no disk components. If M is a product F × I, or if ∂M − F is a set of annuli, then clearly F is not arc-extendible. The main theorem of this paper shows that these are the only obstructions for F to be arc-extendible. Suppose F is a compact incompressible surface on the boundary of a compact, orientable, irreducible 3-manifold M . Let F ′ be a component of ∂M − IntF . We say that F is arc-extendible (in F ) if there is a properly embedded arc γ in F ′ such that F ∪ N(γ) is incompressible. In this case γ is called an extension arc of F . We study the problem of which incompressible surfaces on the boundary M are arcextendible. This is useful in, for example, finding a sequence of mutually nonparallel incompressible surfaces in a 3-manifold. Denote by I the unit interval [0, 1]. We say that M is a product F × I if there is a homeomorphism φ : M ∼= F × I with φ(F ) = F × 1. Note that in this case F ′ = ∂M − IntF , and F is not arc-extendible. A surface F is diskless if it has no disk component. An incompressible surface with a disk component is always arc-extendible, unless the disk lies on a sphere component of ∂M . Thus to avoid trivial cases, we will only consider arc-extension of diskless surfaces. Theorem 1. Let F be a diskless, compact, incompressible surface on the boundary of a compact, orientable, irreducible 3-manifold M , and let F ′ be a non-annular component of ∂M − IntF . Then either F is arc-extendible in F , or M is a product F × I. The proof of the theorem involve some deep results about incompressible surfaces related to Dehn surgery and 2-handle additions. It breaks down into three cases. The 1991 Mathematics Subject Classification. Primary 57N10.. Freedman was partially supported by an NSF grant. Wu’s research at MSRI was supported in part by NSF grant #DMS 9022140. Typeset by AMS-TEX 1 2 MICHAEL FREEDMAN, HUGH HOWARDS AND YING-QING WU case that F ′ is a thrice punctured sphere is treated in Theorem 4, which shows that if the surface obtained by gluing F and F ′ along one of the boundary curve of F ′ is compressible for all the three boundary curves of F , then M must be a product. The second case is that F ′ is parallel into F (see below for definition). A similar result as above holds in this case. Theorem 9 shows that in the remaining case there is an arc γ intersecting some circle C in F ′ at one point, so that all but at most three Dehn twists of γ along C are extension arcs of F . Moreover, in this case the extension arc γ of F can be chosen to have endpoints on any prescribed components of ∂F . See