Lefschetz fibrations, Floer homology and the smooth topology of 4-manifolds

Lefschetz 纤维、Floer 同源性和 4 流形的光滑拓扑

• 批准号: 0904222
• 负责人: Timothy Perutz
• 金额: $ 14.05万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904222&HistoricalAwards=false
• 关键词: Lefschetz; fibrations; Floer; homology; smooth; Lefschetz; fibrations; Floer; homology; smooth

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Auroux-Donaldson-Katzarkov discovered a startlingly simple picture of smooth non-negative-definite 4-manifolds, generalizing Donaldson's interpretation of symplectic 4-manifolds as Lefschetz pencils. They view a 4-manifold, after blowing it up, as the total space of a broken Lefschetz fibration (BLF). In the simplest picture available, the blown-up 4-manifold X is the union of two 4-manifolds X_1 and X_2, each bounding a fibered 3-manifold Y, where X_1 is a Lefschetz fibration over a disc, and X_2 is essentially standard. This proposal focuses on computational and qualitative consequences of this escription. It concerns the Seiberg-Witten theory of broken Lefschetz fibrations and, more particularly, their "Lagrangian matching invariants" - gadgets developed by the P.I. using symplectic geometry associated with BLFs which conjecturally recapture the Seiberg-Witten invariants. The P.I. will use conceptual tools from symplectic topology to compute Floer-theoretic invariants for the two parts of the 4-manifold: X_1 (combinatorially complicated but symplectic) and X_2 (simple but non-symplectic). These computations are directed at making inroads into some major open problems in 4-dimensional topology: existence of symplectic structures, Seiberg-Witten simple type, and algorithmic computation of Seiberg-Witten invariants. They aim to shed light on the algebraic structures of 3- and 4-dimensional gauge theory.Mathematicians regard 4 as the most mysterious dimension - more so than 2, 3, 5 or 1000. It is also the dimension of physical space- time, and equations devised by physicists have led to techniques that probe the topological structure of 4-dimensional spaces and show that they are governed by more complicated rules than anything in higher dimensions. So far, we have a very limited knowledge of what those rules are, and understanding them better is the focus for this project. Recent developments have shown that we can build all 4- dimensional spaces (technically, smooth, compact 4-dimensional manifolds) from simple but highly structured building blocks. This project will study how the known characteristics (invariants) for 4- dimensional manifolds can be understood in terms of those building blocks by invoking methods from another part of geometry that evolved from physics, symplectic topology. One aim is to elucidate what geometric information the invariants capture. Another is to seek genuinely new invariants. One can hope to do so by using the known invariants as a template; but that will require a deep understanding of how those invariants arise from the building blocks.

该奖项是根据2009年《美国复苏与再投资法》（公法111-5）资助的。 Auroux-Donaldson-Katzarkov发现了光滑的非阴性的4个manifolds的惊人简单图片，从而推广了唐纳森（Donaldson）对simplectic 4 manifolds作为lefschetz铅笔的解释。他们将4个manifold炸毁后，是破碎的Lefschetz纤维（BLF）的总空间。 在可用的最简单图片中，吹动的4个manifold X是两个4个manifolds X_1和X_2的结合，每个X的结合在圆盘上是Lefschetz振动，其中X_1是在圆盘上的LEFSCHETZ，而X_2本质上是标准的。该提案着重于该陈述的计算和定性后果。它涉及塞伯格（Seiberg）折断的Lefschetz纤维的理论，尤其是他们的“拉格朗日匹配不变性” - P.I.开发的小工具。使用与BLF相关的符号几何形状，这些几何形状可以猜测将Seiberg-witten的不变性恢复。 P.I.将使用从符号拓扑的概念工具来计算4个manifold的两个部分：x_1（组合复杂但符合性的）和x_2（简单但非及可传播）。这些计算旨在使四维拓扑中的一些主要开放问题陷入困境：符号结构的存在，Seiberg-witten的简单类型和Seiberg-witten不变性算法的算法计算。他们的目的是阐明3维理论和4维理论的代数结构。数学家将4视为最神秘的维度 - 比2、3、5或1000。这也是物理时空的维度，并且由物理学家设计的方程式是对4二维规则的拓扑结构的探讨，而这些方程比任何二维规则都更加严格，以使他们更加严格地构成了他们对他们的拓扑结构的研究。 到目前为止，我们对这些规则的了解非常有限，并且更好地理解它们是该项目的重点。最近的事态发展表明，我们可以从简单但高度结构化的构件中构建所有4维空间（技术，平滑，紧凑的4维歧管）。该项目将研究如何通过从物理学，符号拓扑演变的几何部分中调用几何部分的方法来理解4维流形的已知特征（不变性）。 一个目的是阐明不变的捕获的几何信息。另一个是寻求真正的新不变性。可以希望通过将已知的不变式用作模板来做到。但这将需要深入了解这些不变的构件如何产生。