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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。Auroux-唐纳森-Katzarkov发现了光滑非负定4-流形的一个惊人的简单图像，将唐纳森对辛4-流形的解释推广为莱夫谢茨铅笔。他们认为一个四维流形，在把它吹开之后，是一个破碎的莱夫谢茨纤维化（BLF）的总空间。 在最简单的描述中，爆破的4-流形X是两个4-流形X_1和X_2的并集，每个4-流形X_1和X_2包围一个纤维化的3-流形Y，其中X_1是圆盘上的莱夫谢茨纤维化，X_2本质上是标准的。这个建议的重点是计算和定性的后果，这种描述。它涉及塞伯格-威滕理论的破莱夫谢茨纤维，更具体地说，他们的“拉格朗日匹配不变量”-小工具开发的PI。使用与BLF相关联的辛几何，其精确地重新捕获Seiberg-Witten不变量。私家侦探将使用辛拓扑学的概念工具来计算四维流形的两个部分：X_1（组合复杂但辛）和X_2（简单但非辛）的Floer理论不变量。这些计算的目的是使入侵到一些主要的开放问题在4维拓扑：存在的辛结构，赛伯格-威滕简单型，算法计算赛伯格-威滕不变量。他们的目标是揭示三维和四维规范理论的代数结构。数学家认为4是最神秘的维度-比2，3，5或1000更神秘。它也是物理时空的维度，物理学家设计的方程已经导致了探测四维空间拓扑结构的技术，并表明它们受到比更高维度中任何东西都更复杂的规则的支配。 到目前为止，我们对这些规则的了解非常有限，更好地理解它们是这个项目的重点。最近的发展表明，我们可以从简单但高度结构化的构建块构建所有的四维空间（技术上，光滑，紧凑的四维流形）。这个项目将研究如何通过调用从物理学，辛拓扑学演变而来的几何学的另一部分的方法来理解四维流形的已知特征（不变量）。 一个目的是阐明几何信息的不变量捕获。另一个是寻找真正新的不变量。人们可以希望通过使用已知的不变量作为模板来做到这一点;但这需要深入理解这些不变量是如何从构建块中产生的。

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.