Topology of Three Manifolds

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Proposal: DMS-9971660 Principal Investigator: Marc Culler and Peter Shalen Abstract: Culler and Shalen are proposing to study the set of boundary slopes of a knot, both in the case of an arbitrary closed orientable 3-manifold and in the case of a manifold with cyclic fundamental group. This is a topic of intrinsic interest, which Culler and Shalen believe is also related to some of the most difficult unsolved problems in 3-manifold theory. Their program for applying results about essential surfaces in knot exteriors to 3-manifold theory have led Culler and Shalen to a number of questions about the character variety of a hyperbolic knot group, which they propose to work on. These ideas also lead to an approach to the very broad problem of giving a quantitative version of Thurston's Dehn surgery theorem; this can be seen as the problem underlying most of the existent work on Dehn surgery. The character variety has a distinguished 1-dimensional irreducible component X which is the one that contains the character of the representation that determines the hyperbolic structure on the knot. One question which will be addressed by Culler and Shalen is that of determining properties that distinguish surfaces associated to ideal points of the main component of the character variety from other essential surfaces in the knot exterior. Another question involves the relationship between the hyperbolic volume of the knot complement and properties of a certain factor of the A-polynomial, this factor being the defining equation of a plane curve closely related to X. There is a general notion of hyperbolic volume which applies to a representation whose character lies on X. A third question is that of understanding what properties of an essential surface associated to an ideal point of X guarantee that this generalized volume tends to 0 at the ideal point. Yet another relevant question is that of understanding how often it happens that the curve X is invariant under complex conjugation, and what it means, in terms of the topology of the 3-manfiold, for this to happen. In a somewhat different direction, Culler and Shalen have developed methods for relating the boundary slopes and the genera of essential surfaces in certain situations. They are proposing to develop extensions of this theory. Their techniques in this area may be relevant to the general question of whether every knot in a homotopy sphere has at least one nonzero integer boundary slope.A fundamental problem in many areas of mathematics is to classify all examples of a certain type of mathematical object. The objects of study in this proposal are 3-manifolds, which are mathematical models of 3-dimensional spaces. Since our universe is a 3-dimensional space, the classification of 3-manifolds is directly related to our understanding of nature itself. The classification problem for 3-manifolds is far from solved, but the work of many mathematicians over the last 20 years has at least produced a conjectural answer. The work supported by this problem forms part of the effort to verify the conjectured geometric classification of 3-manifolds.