Surfaces in 3-manifolds

DMS-0203680 Jennifer C. Schultens提出的研究涉及3-流形的研究。 三维流形的概念构成了二维曲面概念的三维类似物。 平面的二维概念应该以一种相当宽泛（也相当专业）的方式来理解。 它包括二维球面（往往被描绘成三维空间中距离原点正好为1的所有点），环面（经常被开玩笑地描述为“甜甜圈上的糖衣”），克莱因瓶，以及许多其他的东西。 三维流形的研究比曲面的研究复杂得多。 曲面是完全分类的，三维流形不是。 事实上，目前还不知道三维流形是否可以分类（在算法意义上）。 已知4-流形不能分类。 本文的研究是在研究Heegaard分裂与Haken分解的关系的基础上，利用三维流形中不同类型的曲面及其相互之间的关系来研究三维流形的结构。 事实证明，在调查中学到的教训有应用到更广泛的问题，在3-流形拓扑，从加性属性的广义桥数的节点和加性属性的宽度的节点的问题，属的Heegaard分裂，这是一个共同的稳定两个给定的Heegaard分裂。 本研究的方法包括M. Scharlemann和P.I.沿着了轨道Heegaard分裂的概念. 这些方法还包括Heegaard分裂的非伸缩性概念、Cerf理论、薄度论证以及对应于Heegaard分裂或Heegaard分裂的非伸缩性的Morse函数所引起的叶理分析。 提出的研究还包括一个策略，以获得更多的结补曲面的结构理论。

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.