Surfaces in 3-manifolds

jennifer C. schultens提议的研究涉及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模拟。二维曲面的概念应该以一种相当广泛（也相当专业）的方式来理解。它包括二维球体（通常被描述为与原点距离正好为1的三维空间中的所有点）、环面（通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物体。3-流形的研究要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，目前尚不清楚3流形是否可以分类（在算法意义上）。众所周知，4流形是不能分类的。本研究试图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。提出的研究源于对heegaard分裂与Haken分解之间关系的研究。事实证明，在该研究中吸取的教训可以应用于更广泛的3流形拓扑问题，从结点的广义桥数的可加性性质和结点宽度的可加性性质到关于Heegaardsplitting的属的问题，即两个给定Heegaardsplitting的共同稳定性。这项研究的方法包括M. Scharlemann和P.I开发的计数技术以及一个轨道的heegard分裂的概念。这些方法进一步包括heegard分裂的不可伸缩的概念、Cerf理论、薄位论证以及与heegard分裂或heegard分裂的不可伸缩相对应的摩尔函数引起的叶状分析。提出的研究还包括一个策略，以获得更多的结构理论的表面在结补。

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.