Incompressible Surfaces in Once‐Punctured Torus Bundles

For the past two decades a large part of the research in the topology of 3-manifolds has been done under the hypothesis that the manifolds are sufficiently-large, that is, they contain properly embedded, incompressible surfaces. The notion of incompressible surface was introduced by W. Haken and the power of this hypothesis was exhibited in the work of F. Waldhausen. Until recently, the only known examples of orientable, irreducible 3-manifolds that are not sufficiently-large were certain 'small' Seifert fibred spaces; the only ones with infinite fundamental group are discussed by Waldhausen in [5]. However, W. Thurston discovered that most Dehn surgeries on the 'figure-eight' knot in S result in orientable, irreducible 3-manifolds that are not sufficiently-large and not Seifert fibred. These new manifolds have infinite fundamental group (in fact, they are hyperbolic). This work has been extended by Hatcher and Thurston to all 2-bridge knots in S [3]. The idea is quite straightforward; namely, if M is obtained from M by doing Dehn surgery along a simple closed curve k in M and M contains an orientable, incompressible surface, then the bounded manifold M' = M — u{k), where u(k) is an open tubular neighbourhood of k, contains a properly-embedded, orientable, incompressible and boundary-incompressible surface. The problem is, therefore, to understand the incompressible and boundaryincompressible surfaces in M'. This problem is, in its own right, extremely important to the understanding of the structure of 3-manifolds.