Gauge theory, homology cobordisms, and Rohlin's invariant

规范场论、同调配边主义和罗林不变量

• 批准号: 0505605
• 负责人: Daniel Ruberman
• 金额: $ 17.77万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505605&HistoricalAwards=false
• 关键词: Gauge; theory; homology; cobordisms; Rohlin

Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten and Yang-Mills gauge theory. The first part of the project, to be carried out in collaboration with Nikolai Saveliev and Tomasz Mrowka, is to understand the smooth topology of 4-manifolds that homologically resemble a product of a 3-dimensional manifold with a circle. The central questions center around the interpretation of the classical Rohlin invariant in terms of gauge theory; an important facet is the analysis of end-periodic Dirac operators. Solutions of the main problems posed will decide the existence of some manifolds predicted by high-dimensional topology (surgery theory) and settle a fundamental question about the homology cobordism group. This latter question is key in understanding the triangulability of manifolds of dimension 5 or more. The second part of the project proposes joint work with Saso Strle. We plan to investigate the degree to which invariants of a 3-manifold derived from the Seiberg-Witten equations constrain the topology of those 4-manifolds which it bounds. The results will be used to investigate a classical problem about group actions on the K3 surface and other algebraic varieties. Many of the most difficult problems in topology are concerned with the structure of three and four-dimensional manifolds. Questions about phenomena in these dimensions have particular interest because those are the dimension of space and time that make up our world. The research to be carried out under this grant makes use of powerful geometric and analytical techniques that have been introduced into the field over the last twenty years, under the general rubric of gauge theory. Much of the application of four-dimensional gauge theory has been to manifolds that are simply-connected, in which any loop can be shrunk to a point. The research proposed will use Yang-Mills and Seiberg-Witten gauge theories to shed light on manifolds which are not simply-connected, where new phenomena are expected. By studying the Dirac operator (originally introduced to give a relativistic quantum theory of the electron) on manifolds with a certain type of periodicity, we seek to get restrictions on an invariant of non-simply connected manifolds, called the Rohlin invariant. This subtle invariant is connected to three-dimensional topology, and its behavior determines whether high-dimensional manifolds may be triangulated.

丹尼尔·鲁伯曼将利用Seiberg-Witten和Yang-Mills规范理论进行几何拓扑学研究。该项目的第一部分将与Nikolai Saveliev和Tomasz Mrowka合作进行，旨在了解4维流形的光滑拓扑，这些流形同源类似于3维流形与圆的乘积。中心问题围绕着规范理论对经典罗林不变量的解释；一个重要的方面是对末周期狄拉克算符的分析。所提出的主要问题的解将决定高维拓扑(外科理论)所预测的某些流形的存在，并解决关于同调余边群的一个基本问题。后一个问题是理解5维或更多维流形的三角性的关键。该项目的第二部分建议与Saso Strle合作。我们计划研究从Seiberg-Witten方程导出的3-流形的不变量对它所界定的4-流形的拓扑的约束程度。这些结果将被用来研究关于K3曲面和其他代数簇上的群作用的一个经典问题。拓扑学中的许多最困难的问题都与三维和四维流形的结构有关。关于这些维度中的现象的问题特别令人感兴趣，因为这些维度是构成我们世界的空间和时间维度。在这项资助下进行的研究利用了过去二十年来在规范理论的一般范畴下引入该领域的强大的几何和分析技术。四维规范理论的大部分应用是在简单连接的流形上，在这种流形中，任何回路都可以收缩到一个点。提出的研究将使用杨-米尔斯和塞贝格-威腾规范理论来阐明不是简单联系的流形，这些流形有望出现新的现象。通过研究具有一定周期的流形上的Dirac算符(最初是为了给出电子的相对论量子理论)，我们试图得到对非单连通流形的一个不变量的限制，称为Rohlin不变量。这种微妙的不变量与三维拓扑有关，它的行为决定了高维流形是否可以三角剖分。