Mathematical Sciences: Essential Laminations in 3-Manifolds

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9400651 Brittenham Essential laminations generalize two objects important to 3-manifold topology: the incompressible surface and the taut foliation. Both of these more `classical' objects have proved themselves to be very useful in the past (and continue to be so, today), and essential laminations have recently shown similar power in attacking many of the fundamental problems in the theory of 3-manifolds. The investigator intends to continue his work on essential laminations, with three main goals in mind: (1) to show that they behave in many situations much like their far more manageable cousin, the incompressible surface, (2) to show that homotopy equivalent 3-manifolds, which contain essential laminations, are homeomorphic, and (3) to construct essential laminations in a wide variety of 3-manifolds, using the notion of Haken normal form for laminations. In other words, the investigator intends to show that essential laminations (1) behave nicely, (2) tell us interesting things about 3-manifolds, and (3) can be found in `most' 3-manifolds. It is somehow surprising that, in spite of the fact that we live in a 3-dimensional world, and therefore have a great deal of natural intuition about how things `work' in 3 dimensions, our understanding of 3-dimensional manifolds, objects modelled on our 3-dimensional space, is far from complete. Many of its basic problems, such as the celebrated Poincare conjecture, remain unsolved, even though the analogous problems in higher dimensions have been settled. Many new techniques have been developed, and continue to be developed, to try to unravel this familiar, though mysterious, dimension. The investigator intends to continue to delevop a new technique for studying 3-dimensional manifolds which uses (well-understood) 2-dimensional objects to study these (not so well-understood) 3-dimensional ones. The idea is that by thinking of a 3-dimensional object as being made up of a collection of 2-dimensionsal `sheets' , stacked together, we can sometimes use our (even better!) intuition and understanding about dimension 2 to `stitch together' a solution to a 3-dimensional problem. ***