Exploring problems in 3- and (3+1)-dimensional topology

总的来说，Scharlemann的研究集中在经典结和3流形的理论上，特别是在使用heeggaard分裂和经典结理论的相关概念上。重点是3-流形中包含的曲面的行为，但以现代的观点：将图论的思想添加到关于曲面相交的老式组合论证中；将缝合流形理论添加到经典的3流形层次概念中（因此参数化曲面和Thurston范数的估计有助于控制和理解拓扑）；并在莫尔斯理论的经典工具中添加了薄位置，这样临界指数的手柄就不会一次全部添加，而是尽可能慢地添加。最近，Scharlemann将这些和类似的新工具应用于解决4流形拓扑中的问题，最突出的是Schoenflies猜想，以及关于经典连杆之间的联系的长期问题，它们在4球中的可能切片，以及手术如何影响两者。例如，最近与Abby Thompson和Bob Gompf的合作使用了缝合流形和Heegaard理论来寻找广义性质R猜想的最简单的反例。我们证明了有一个2组分可能的反例，其中包含一个方结（或者，可能看起来更容易，一个1属结，到目前为止尚未确定）作为一个组分。这些和其他结果可能是将3维流形思想有用地应用于那些有趣的拓扑问题的第一步，这些问题有时被称为3 + 1维，因为它们询问3维和4维流形是如何相互关联的。对我们周围世界最基本的观察之一就是它有三维空间。这个提议的广泛意图是帮助我们更好地理解具有这种特性的空间：任何生活在其中的人都会看到他们的世界是这样的，即三维的。除了模拟我们周围的宇宙之外，这些被称为“3流形”的空间还支持一些有趣的现象，比如打结，在打结中，一个简单的物体，比如一根花园水管（或者一串DNA，或者更一般的，一个化学分子）可以以一种复杂的方式存在于空间中。这些3-流形也位于具有额外深刻物理意义的空间边缘，这些空间在局部看起来是4维空间（如物理学家通常所说的时空）。我们对三维流形的新兴理解也可以帮助我们理解关于这些四维流形的长期问题。本提案特别强调的是位于三维和四维流形理论之间的问题，并结合了两者的打结现象。这些维度之所以有趣，部分是因为它们模拟了我们生活的宇宙；在它们之间打结是一种加深我们对这些空间之间关系的理解的工具。

In general, Scharlemann's research has centered on the theory of classical knots and of 3-manifolds, in particular on the use of Heegaard splittings and of related notions from classical knot theory. The focus has been on the behavior of surfaces contained in the 3-manifolds, but with a modern viewpoint: Add ideas from graph theory to old-fashioned combinatorial arguments on surface intersections; add sutured manifold theory to the classical notion of hierarchies on 3-manifolds (so parameterizing surfaces and estimates of the Thurston norm help control and understand the topology); and add thin position to the classic tools of Morse theory, so handles of the critical index are added not all at once, but as slowly as possible. Recently Scharlemann has applied these and similar new tools towards resolving questions in the topology of 4-manifolds, most prominently the Schoenflies Conjecture, but also long-standing questions about the connection between classical links, their possible slicings in the 4-ball, and how surgery can affect both. For example, recent work with Abby Thompson and Bob Gompf uses sutured manifold and Heegaard theory to search for the easiest possible counterexample to the Generalized Property R Conjecture. We show that there is a 2-component likely counterexample containing a square knot (or, maybe seen as easier, a genus 1 knot, so far undetermined) as a component. These and other results may be the first steps towards a useful application of 3-manifold ideas to those intriguing topological questions which are sometimes called 3 + 1-dimensional because they ask how 3- and 4-dimensional manifolds are interrelated. One of the most basic observations about the world around us is that it has three dimensions. The broad intent of this proposal is to help us better understand spaces with precisely this property: anyone living in one would see their world as of this sort, namely 3-dimensional. Beyond modeling the universe around us, these spaces, called "3-manifolds", support interesting phenomena such as knotting, in which a simple object like a garden-hose (or a string of DNA, or, more generally, a chemical molecule) can lie in space in a complicated way. These 3-manifolds also lie on the edge of spaces with additional deep physical significance, those that locally appear to be 4-dimensional spaces (such as physicists often refer to as space-time). Our emerging understanding of 3-dimensional manifolds can also help us understand long-standing questions about these 4-dimensional manifolds. The particular emphasis in this proposal is on questions that sit on the edge between 3- and 4-dimensional manifold theory and incorporate knotting phenomena from both. The dimensions are interesting in part because they model the universe in which we live; knotting within them is a tool to deepen our understanding of how such spaces relate.