Surfaces in 3-manifolds

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DMS-0203680Jennifer C. SchultensThe proposed research concerns the study of 3-manifolds. The notionof a 3-manifold constitutes the 3-dimensional analogue of the2-dimensional notion of a surface. The 2-dimensional notion ofsurface is to be understood in a rather broad (and rather technical)way. It includes the 2-dimensional sphere (that tends to be picturedas all points in 3-space at distance exactly 1 from the origin), thetorus (often, jokingly, described as ``the icing on a doughnut''), theKlein bottle, and many others. The study of 3-manifolds isconsiderably more complex than that of surfaces. Surfaces arecompletely classified, 3-manifolds are not. In fact, it is at presentunknown whether 3-manifolds can be classified (in an algorithmicsense). It is known that 4-manifolds cannot be classified. Theresearch here endeavors to employ different types of surfaces lying in3-manifolds and their relation to each other in a structural study of3-manifolds.The proposed research grows out of a study of the relation of Heegaardsplittings to Haken decompositions. It turns out that the lessonslearnt in that investigation have applications to a wider range ofproblems in 3-manifold topology ranging from additivity properties ofthe generalized bridge numbers of knots and additivity properties ofthe width of knots to questions about the genus of a Heegaardsplitting that is a common stabilization of two given Heegaardsplittings. The methods for this investigation include a countingtechnique developed by M. Scharlemann and the P.I along with thenotion of an orbifold Heegaard splitting. The methods further includethe notion of untelescoping of Heegaard splittings, Cerf theory, thinposition arguments, and the analysis of foliations induced by Morsefunctions corresponding to Heegaard splittings or untelescopings ofHeegaard splittings. The proposed research also includes a strategyto obtain a more structural theory of surfaces in knot complements.