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)提供资金的。Auroux-Donaldson-Katzarkov发现了一个非常简单的光滑非负定4-流形的图景，将Donaldson对辛4-流形的解释推广为Lefschetz铅笔。他们认为一个4维流形在爆炸后，是一个破碎的Lefschetz纤维(BLF)的总空间。在现有的最简单的图中，爆破的4-流形X是两个4-流形X_1和X_2的并，每个X_1和X_2都有一个纤维状的3-流形Y，其中X_1是圆盘上的Lefschetz纤维，而X_2本质上是标准的。这项提案侧重于这一eScription的计算和质量后果。它涉及Seiberg-Witten关于断裂的Lefschetz纤维的理论，更具体地说，他们的“拉格朗日匹配不变量”--P.I.使用与BFL相关的辛几何开发的小工具，推测地重新捕获了Seiberg-Witten不变量。P.I.将使用辛拓扑中的概念工具来计算4-流形的两个部分的Floer理论不变量：X_1(组合复杂但辛)和X_2(简单但非辛)。这些计算旨在解决四维拓扑学中的一些主要公开问题：辛结构的存在性、Seiberg-Witten简单类型和Seiberg-Witten不变量的算法计算。他们的目标是阐明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导致了探索4维空间拓扑结构的技术，并表明它们受到比更高维度的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，更好地理解它们是本项目的重点。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4维空间(从技术上讲，平滑、紧凑的4维流形)。这个项目将研究如何通过引用从物理-辛拓扑演变而来的几何学的另一部分的方法，来根据这些构件来理解4维流形的已知特征(不变量)。一个目的是阐明不变量捕捉到了什么几何信息。另一种是寻找真正的新不变量。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.