Curves and 3-Manifolds

我们在这个建议中描述了三个问题，都是在纽结理论和3-流形的一般领域。3-流形的Heegaard分裂是将3-流形分解成简单的片断，称为HandleBody。这种分解是研究关于3-流形的许多有趣的公开问题的有效方法，但在我们关于Heegaard分裂的基本信息中有一些惊人的空白。我们知道，给定同一三维流形的两个Heegaard分裂，其中一个可以通过一系列运动从一个移动到另一个。这个方案中的第一个问题是，在这些移动下，一个3-流形的两个不同的Heegaard分裂可以有多远。第二个问题是探索彭加莱猜想的一个特定方面，该猜想是由邓伍迪最近的尝试以及弗里德曼和邱腾华的早期工作提出的。我们将仔细研究是否可以使一个类似邓伍迪提出的计划在一个真正的三球上起作用，首先从一个更简单的版本的三角两球面上的问题开始。最后一个问题是作者最近关于平面中浸没曲线到三维空间中浸没2球面的工作的推广。我们将研究奇点集(例如，包括自交曲线和零曲率点)，并尝试推导奇点数和奇点类型之间的关系，类似于平面或射影2-空间中浸入曲线的已知结果。我们还描述了提出者如何使用这些问题和其他相关问题来激发从高中生到博士后从高中生到本科生和研究生的对该领域和一般数学的兴趣。低维拓扑是研究空间在二、三和四维空间的性质的学科。这是一个过去完全属于“纯”数学领域的领域，这一领域的研究主要是因为它的复杂性和美感而受到追捧。随着我们继续了解宇宙，从空间本身的形状到DNA链的打结，这个抽象领域与现实世界之间的深层联系日益明显。这一提议旨在探索三维空间和纽结理论中的一些基本问题，包括理解三维空间不同分解之间的关系，以及量化沉浸在标准三维宇宙中的二维球体的复杂性。

We describe three problems in this proposal, all in the general area of knot theory and 3-manifolds. A Heegaard splitting of a 3-manifold is a decomposition of the 3-manifold into simple pieces, called handlebodies. This decomposition is an effective way to study many of the interesting open questions about 3-manifolds, but there are some startling gaps in our basic information about Heegaard splittings. We know that given two Heegaard splittings of the same 3-manifold, one can move from one to the other by a series of moves. The first problem in this proposal asks how "far apart" two different Heegaard splittings of a 3-manifold can be under these moves. The second problem is to explore a particular aspect of the Poincare Conjecture prompted by Dunwoody's recent attempt and suggested by earlier work of Freedman and Yau. We will look carefully at whether a plan analogous to what Dunwoody proposed can be made to work on a real 3-ball, beginning with an even simpler version of the question on a triangulated 2-sphere. The final problem is a generalization of recent work of the proposer on immersed curves in the plane to immersed 2-spheres in 3-space. We will look at the set of singularities (including, for example, curves of self-intersection and points of zero curvature) and try to derive relationship between numbers and types of singularities, similar to the kinds of results known for immersed curves in the plane or in projective 2-space. We also describe how these and other related problems have been used by the proposer to stimulate interest in the field and in mathematics in general at levels ranging from high school students, including undergraduates and graduate students, through postdocs.Low-dimensional topology is the study of properties of spaces in dimensions two, three, and four. It is a field that used to lie squarely in the realm of "pure" mathematics, and research in the field was pursued largely for its intricacy and beauty. As we continue to understand the universe, from the shape of space itself to the knotting of strands of DNA, the deep connections between this abstract area and the real world are increasingly apparent. This proposal aims to explore some of the fundamental questions in 3-dimensional spaces and knot theory, including understanding the relationships between different decompositions of 3-dimensional spaces and quantifying the complexity of 2-dimensional spheres immersed in a standard 3-dimensional universe.