Gauge theory, homology cobordisms, and Rohlin's invariant

丹尼尔鲁伯曼将开展研究的几何拓扑结构，使用塞伯格-威滕和杨米尔斯规范理论。该项目的第一部分，将与Nikolai Saveliev和Tomasz Mrowka合作进行，是了解4-流形的光滑拓扑，这些流形同调类似于三维流形与圆的乘积。 中心问题集中在规范理论中经典的罗林不变量的解释上;一个重要的方面是对周期性狄拉克算符的分析。所提出的主要问题的解决方案将决定存在的高维拓扑（手术理论）预测的一些流形，并解决一个基本问题的同调配边群。 后一个问题是理解5维或更高维流形的可三角化性的关键。 该项目的第二部分建议与Saso Strle联合开展工作。我们计划研究从Seiberg-Witten方程推导出的3-流形的不变量对它所界定的4-流形的拓扑结构的约束程度。 这些结果将被用来研究K3曲面上的群作用和其他代数簇的经典问题。 拓扑学中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有趣，因为这些是构成我们世界的空间和时间的维度。 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标题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都是在单连通的流形上，其中任何环都可以收缩为一点。 这项研究将使用杨-米尔斯和塞伯格-威滕规范理论来阐明非简单连通的流形，预计会出现新的现象。 通过研究狄拉克算子（最初是为了给出电子的相对论量子理论而引入的）在具有某种类型的周期性的流形上，我们试图得到对非单连通流形的不变量的限制，称为Rohlin不变量。 这个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。

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.