Surfaces in 3-manifolds

DMS-0203680 Jennifer C.Schultens拟议的研究涉及3-流形的研究。三维流形的概念构成了曲面的二维概念的三维类比。表面的二维概念应该以相当宽泛的(和相当技术性的)方式来理解。它包括2维球体(倾向于描绘成与原点恰好1距离的3维空间中的所有点)、圆角(通常开玩笑地被描述为“甜甜圈上的糖霜”)、克莱因瓶，以及许多其他东西。3-流形的研究比曲面的研究复杂得多。曲面是完全分类的，而3-流形不是。事实上，目前还不清楚3-流形是否可以被分类(在算法意义上)。众所周知，4-流形是不能分类的。本文致力于在三维流形的结构研究中使用位于三维流形中的不同类型的曲面以及它们之间的相互关系。结果表明，在这项研究中所学到的教训适用于三维流形拓扑中的更广泛的问题，从广义纽结桥数的可加性性质和纽结宽度的可加性性质到作为两个给定Heegard分裂的公共稳定化的Heegard分裂的亏格问题。这项研究的方法包括M.Scharlemann和P.I发展的计数技术，以及Orbilold Heegaard分裂的运动。这些方法还包括Heegaard分裂的不可缩合概念、Cerf理论、薄位置论元，以及Heegaard分裂或Heegaard分裂的不可缩合所对应的Morse函数所引起的叶的分析。提出的研究还包括一种策略，以获得纽结补中曲面的更具结构性的理论。

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.