Stein surfaces, 4-manifolds and symplectic topology

斯坦因面、4 流形和辛拓扑

• 批准号: 0603958
• 负责人: Robert Gompf
• 金额: $ 25.52万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603958&HistoricalAwards=false
• 关键词: Stein; surfaces; manifolds; symplectic; topology

DMS-0603958Robert GompfA primary aim of this project is to use topological methods to study which open subsets of complex surfaces are Stein. Preliminary results show that up to topological isotopy, Stein open subsets are as common as possible. That is, if U satisfies the basic necessary condition (it is homeomorphic to an open handlebody with handles of index at most 2), then it is topologically isotopic (not necessarily smoothly or ambiently) to a Stein open subset. The project is revealing much more detailed structure: a 2-complex topological spine K for U that is smoothly embedded except for one point on each 2-cell, and a neighborhood system of K consisting of homeomorphic Stein surfaces indexed by a Cantor set and frequently realizing uncountably many diffeomorphism types. There is much control over minimal genera of homology classes of the Stein neighborhoods. The project investigates these phenomena using techniques from topological 4-manifold theory (Casson handles and reimbedding), and also applies these methods to study pseudoconvex smooth and topological embeddings of 3-manifolds into complex surfaces. A second thrust of the proposal concerns the topology of closed, symplectic 4-manifolds, such as the recently discovered simply connected 4-manifolds with small Euler characteristics. For example, what range of signature and Euler characteristic is realized for a fixed fundamental group? Which symplectic manifolds decompose completely after connected sum with a single CP^2? Can one find general bounds on the number of CP^2 summands required to decompose symplectic manifolds? These questions can be studied via Kirby calculus and Lefschetz fibrations. One can also drop the symplectic structures and ask the above questions for irreducible smooth 4-manifolds.The project concerns the classification of 3- and 4-manifolds and their contact, symplectic and Stein structures. An n-manifold is a space locally indistinguishable from Euclidean n-space. For example the space in which we live and the universe (space-time) are 3- and 4-manifolds, respectively. Contact and symplectic structures first arose in classical physics (optics and Hamiltonian mechanics, respectively), but they are also intimately connected with more recent physics such as string theory, as well as various branches of mathematics such as algebraic geometry, topology and dynamical systems. Stein manifolds are symplectic manifolds endowed with additional structure that is naturally expressed via contact topology. They also have various equivalent definitions in terms of complex analysis - for example, Stein manifolds are essentially the same as complex submanifolds of complex n-space. Complex analysts have recognized Stein manifolds as fundamental objects of study for most of the past century, but much about them remains unknown. The present project investigates which open subsets of a complex manifold are Stein manifolds. This question (like many in topology) is most subtle in the context of 4-manifolds. In this setting, the project has already shown that every subset satisfying the most basic requirements can be deformed into a Stein subset, although the deformation cannot usually be made smooth. Such unsmoothable deformations can only be accessed via esoteric techniques from topological 4-manifold theory. Since these techniques are quite alien to complex analysts, the research is in some sense interdisciplinary, and the results are completely unlike anything obtainable by traditional analytic methods. For example, the resulting Stein manifolds frequently come in uncountably many types, all topologically equivalent but smoothly distinct. In addition to Stein manifolds, the project deals with 4-manifolds and symplectic and contact manifolds. Most of what is known about the classifications of these objects has been discovered in the past few decades. While the fields are developing rapidly, the final classification schemes remain elusive.

DMS-0603958Robert Gompf本项目的主要目的是使用拓扑方法来研究复杂曲面的哪些开子集是Stein。初步结果表明，直到拓扑合痕，Stein开子集尽可能常见。也就是说，如果U满足基本的必要条件（它同胚于一个柄数至多为2的开闭体），那么它拓扑上（不一定是光滑的或环境的）与一个Stein开子集同构。该项目揭示了更详细的结构：一个2-复的拓扑脊K为U，除了每个2-细胞上的一个点外，它是平滑嵌入的，K的邻域系统由康托集索引的同胚Stein曲面组成，经常实现无数的同胚类型。有很大的控制最小属的同源类的斯坦社区。该项目使用拓扑4-流形理论（Casson句柄和重新嵌入）的技术来研究这些现象，并将这些方法应用于研究3-流形到复杂曲面中的伪凸光滑和拓扑嵌入。该建议的第二个推力涉及闭合的辛4-流形的拓扑，例如最近发现的具有小欧拉特征的单连通4-流形。例如，对于一个固定的基本群，可以实现什么范围的签名和欧拉特征？哪些辛流形在与单个CP连接和之后完全分解^[2]？我们能找到分解辛流形所需的CP^2被加数的一般界吗？这些问题可以通过Kirby演算和Lefschetz纤维化来研究。我们也可以放弃辛结构，而对不可约光滑4-流形提出上述问题。该项目涉及3-和4-流形的分类及其接触，辛和Stein结构。一个n-流形是一个与欧几里德n-空间局部不可区分的空间。例如，我们生活的空间和宇宙（时空）分别是3-流形和4-流形。接触和辛结构最早出现在经典物理学（分别是光学和哈密顿力学）中，但它们也与最近的物理学（如弦理论）以及数学的各个分支（如代数几何，拓扑学和动力系统）密切相关。Stein流形是辛流形，它被赋予了通过接触拓扑自然表达的附加结构。它们在复分析中也有各种等价的定义-例如，斯坦流形本质上与复n-空间的复子流形相同。复杂的分析家已经认识到斯坦流形的基本对象的研究在过去的大部分世纪，但他们仍然是未知的。本项目研究复流形的哪些开子集是Stein流形。这个问题（像拓扑学中的许多问题一样）在四维流形的上下文中是最微妙的。在这种情况下，该项目已经表明，满足最基本要求的每个子集都可以变形为Stein子集，尽管变形通常不能平滑。这种不可平滑的变形只能通过拓扑4-流形理论的深奥技术来访问。由于这些技术对复杂的分析师来说是相当陌生的，因此研究在某种意义上是跨学科的，其结果完全不同于传统分析方法所能获得的任何结果。例如，由此产生的斯坦流形经常有无数种类型，所有拓扑等价但光滑不同。除了Stein流形，该项目还涉及4-流形，辛流形和接触流形。关于这些天体的分类，我们所知道的大部分都是在过去几十年中发现的。虽然这些领域正在迅速发展，但最终的分类方案仍然难以捉摸。