Surfaces in 3-manifolds

University Of California-Davis, Davis CA

Investigators

Abstract

DMS-0203680 Jennifer C. Schultens The proposed research concerns the study of 3-manifolds. The notion of a 3-manifold constitutes the 3-dimensional analogue of the 2-dimensional notion of a surface. The 2-dimensional notion of surface is to be understood in a rather broad (and rather technical) way. It includes the 2-dimensional sphere (that tends to be pictured as all points in 3-space at distance exactly 1 from the origin), the torus (often, jokingly, described as ``the icing on a doughnut''), the Klein bottle, and many others. The study of 3-manifolds is considerably more complex than that of surfaces. Surfaces are completely classified, 3-manifolds are not. In fact, it is at present unknown whether 3-manifolds can be classified (in an algorithmic sense). It is known that 4-manifolds cannot be classified. The research here endeavors to employ different types of surfaces lying in 3-manifolds and their relation to each other in a structural study of 3-manifolds. The proposed research grows out of a study of the relation of Heegaard splittings to Haken decompositions. It turns out that the lessons learnt in that investigation have applications to a wider range of problems in 3-manifold topology ranging from additivity properties of the generalized bridge numbers of knots and additivity properties of the width of knots to questions about the genus of a Heegaard splitting that is a common stabilization of two given Heegaard splittings. The methods for this investigation include a counting technique developed by M. Scharlemann and the P.I along with the notion of an orbifold Heegaard splitting. The methods further include the notion of untelescoping of Heegaard splittings, Cerf theory, thin position arguments, and the analysis of foliations induced by Morse functions corresponding to Heegaard splittings or untelescopings of Heegaard splittings. The proposed research also includes a strategy to obtain a more structural theory of surfaces in knot complements.