Low-Dimensional Contact and Symplectic Topology

低维接触和辛拓扑

• 批准号: 1309212
• 负责人: Thomas Mark
• 金额: $ 14.13万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309212&HistoricalAwards=false
• 关键词: Low; Dimensional; Contact; Symplectic; Topology

The topology of smooth 4-dimensional manifolds presents an ever-deepening mystery. It is natural to impose geometric constraints to narrow the possibilities, and an interesting such constraint is the existence of a symplectic structure. In the case of 4-manifolds with boundary the natural structure arising on the boundary is a contact structure; there are many deep and fascinating questions involving contact and symplectic topology, and their interplay with smooth topology coming from gauge-theoretic invariants. The PI will use symplectic and contact geometry in conjunction with Floer homology invariants to study several related questions. First, the PI has introduced several new geometric operations on 4-manifolds that can be realized as instances of monodromy substitutions in Lefschetz fibrations, and has proved convexity results that mean such substitution operations are symplectic. The PI will pursue this line of inquiry with the goal of finding new constructions of exotic symplectic 4-manifolds. Second, the PI will study Stein fillings of contact structures supported by planar open books: by translating the classification problem for Stein fillings into a certain symplectic isotopy problem, the PI will adapt pseudo-holomorphic techniques of Siebert--Tian and others to study finiteness questions for this situation. Next, the PI will study the problem of "symplectic fiber sum decomposition," which is the question of whether a symplectic manifold diffeomorphic to a fiber sum can be realized as a symplectic fiber sum. This question has bearing on the hypothesis that gauge-theoretic invariants do not provide complete information on the diffeomorphism types of 4-manifolds in a fixed homeomorphism class. Finally, he will pursue recent work relating Heegaard Floer homology with an invariant of open book decompositions of 3-manifolds called the fractional Dehn twist coefficient. Since Einstein's description of mechanics and electrodynamics as inherently a four-dimensional theory, the observed universe has generally been conceived as a smooth four-dimensional manifold: that is, a four-dimensional analog of a smooth surface such as a plane or sphere. A fundamental question is then: what manifold is it? To pose a simplified analogy, the surface of the earth is generally "flat" when viewed by a casual observer, but it is a mistake to infer that it is planar. The universe is similarly "mostly flat" on an appropriate distance scale, but its global topology or "shape" is not known. A goal of the 4-manifold topologist, then, is to describe the list of possibilities for the structure of the universe, in analogy with the relatively easy-to-understand list of possible surfaces from which a "generally flat" object like the surface of the earth can select. There are two ways to focus this effort: by introducing geometric structures (in this case a "symplectic" structure, loosely a way of measuring area or volume) that do not exist on every 4-manifold, yet leave a rich and interesting class of examples to study and may also give a stepping-stone to the general situation; and by "cutting" a complicated manifold into simpler pieces and studying the latter individually. The work supported by this grant will use both these approaches: apply techniques recently introduced by the PI and collaborators to construct interesting new examples of symplectic 4-manifolds; and also study whether and how symplectic 4-manifolds can be decomposed into simpler pieces. Moreover, the PI hopes to make progress in classifying some of the most rigid of these pieces, known as Stein manifolds, with the aim of exploring some of the poorly-understood myriad of possibilities in smooth 4-dimensional topology.

光滑四维流形的拓扑结构呈现出一个不断加深的奥秘。很自然地，要施加几何约束来缩小可能性，而一个有趣的约束就是辛结构的存在。在具有边界的4-流形的情况下，边界上产生的自然结构是接触结构;有许多涉及接触和辛拓扑的深刻而迷人的问题，以及它们与来自规范论不变量的光滑拓扑的相互作用。PI将使用辛和接触几何与Floer同调不变量一起研究几个相关问题。首先，PI在4-流形上引入了几种新的几何运算，这些运算可以在Lefschetz纤维化中实现为单值替换的实例，并证明了凸性结果，这意味着这种替换运算是辛的。PI将继续这一研究方向，目标是找到奇异辛4-流形的新构造。其次，PI将研究平面开卷支撑的接触结构的Stein填充：通过将Stein填充的分类问题转化为一定的辛合痕问题，PI将采用Siebert-Tian等人的伪全纯技术来研究这种情况下的有限性问题。接下来，PI将研究“辛纤维和分解”的问题，这是一个辛流形是否可以实现为辛纤维和的问题。这个问题与规范理论不变量不能提供关于固定同胚类中4-流形的同胚类型的完整信息的假设有关。最后，他将追求最近的工作有关Heegaard Floer同源性与不变量的开卷分解的3流形称为分数德恩扭曲系数。由于爱因斯坦将力学和电动力学描述为固有的四维理论，所观察到的宇宙通常被认为是一个光滑的四维流形：也就是说，一个光滑表面（如平面或球体）的四维模拟。一个基本的问题是：它是什么流形？做一个简单的类比，当一个不经意的观察者看到地球表面时，地球表面通常是“平的”，但推断它是平面的是错误的。在适当的距离尺度上，宇宙同样是“大部分平坦的”，但它的整体拓扑或“形状”尚不清楚。因此，四维拓扑学家的一个目标是描述宇宙结构的可能性列表，类似于相对容易理解的可能表面列表，像地球表面这样的“一般平坦”物体可以从中选择。有两种方法来集中这项工作：通过引入几何结构（在这种情况下是一种“辛”结构，松散地测量面积或体积的方法），这些结构不存在于每个4-流形上，但留下了丰富而有趣的一类例子来研究，也可以为一般情况提供一块垫脚石;通过“切割”一个复杂的流形成更简单的碎片，并单独研究后者。这项资助支持的工作将使用这两种方法：应用PI和合作者最近引入的技术来构建辛4-流形的有趣的新例子;并研究辛4-流形是否以及如何分解成更简单的部分。此外，PI希望在分类这些最刚性的部分方面取得进展，这些部分被称为Stein流形，目的是探索光滑四维拓扑中一些难以理解的无数可能性。