Equivariant Gauge Theory on 3-Manifolds

提案：DMS-0071480 PI：Nikolai Saveliev使用Floer同调来研究三维流形的一个主要障碍是缺乏对其行为的清晰理解，相对于三维拓扑的标准结构，如Dehn手术，连接和等，在这个方向上的已知结果在Floer同调计算或应用中没有取得多大进展。研究者的主要论点是，Floer同调最自然地表现的操作是分支覆盖。因此，研究者研究了群作用三维流形上的规范理论，并将规范理论不变量与分支集的经典不变量联系起来。作为一个例子，他以前的研究等变规范理论的联系准齐次表面奇点导致了一个封闭的形式公式表达其弗洛尔同源性方面的经典不变量。引入并研究了具有循环群作用的积分同调球面的等变Casson不变量，并将其与经典Casson不变量和纽结签名联系起来。该定义利用了等变规范理论，特别是等变Floer指数。在反全纯对合的情况下，后者与Stein曲面的拓扑和非紧Kaehler几何密切相关。调查员还继续他的研究的联系completecutive奇点，和相关的等变卡森不变的同源cobordisms和三角测量猜想拓扑流形在五个维度和更高。本研究是从理论物理学出发，用规范理论的方法研究三维和四维流形。在过去的二十年里，这些方法为经典的低维拓扑学带来了新的生命，导致了许多引人注目的发展和许多困难问题的解决。这个领域发生了革命性的变化，以至于二十年前的知识状态在许多方面看起来像是一片近乎完全无知的沙漠。四维流形是受益最大的流形：规范理论方法在这种情况下非常有效。与此同时，他们在三维中的类似物，也就是所谓的弗洛尔同源性，还没有完全展现出它的潜力。Floer同源性是研究人员研究工作的重点。

Proposal: DMS-0071480PI: Nikolai SavelievA major obstruction to using the Floer homology to study 3-dimensional manifolds is the lack of clear understanding of its behavior with respect to the standard constructions of 3-dimensional topology, such as Dehn surgery, connected sums etc. The known results in this direction have not made much of headway in Floer homology computations or applications. The main thesis of the investigator is that the operation with respect to which the Floer homologybehaves most naturally is the branched covering. Therefore, the investigator studies gauge theory on 3-manifolds with group actions and relates gauge-theoretical invariants to the classical invariants of the branch set. As an example, his previous study of equivariant gauge theory on the links of quasi-homogeneous surface singularities has led to a closed form formula expressing their Floer homology in terms of classical invariants. The investigator introduces and studies an equivariant Casson invariant for an integral homology sphere with a cyclic group action and relates it to the classical Casson invariant and the knot signatures. The definition makes use of the equivariant gauge theory and, in particular, of the equivariant Floer index. The latter is closely related, in the case of anti-holomorphic involutions, to the topology of Stein surfaces and the non-compact Kaehler geometry. The investigator also continues his study of the links of completeintersection singularities, and the relevance of the equivariant Casson invariant to homology cobordisms and the triangulation conjecture for topological manifolds in dimensions five and higher. The research is a study of 3- and 4-dimensional manifolds by the methods of gauge theory, coming from theoretical physics. In the last two decades, these methods brought new life into the classical low dimensional topology, leading to many spectacular developments and to the solutions of many difficult problems. The area was revolutionized to the point that the state of knowledgeof two decades ago looks now, in many aspects, like a desert of nearly complete ignorance. Manifolds of dimension four were the ones that benefited the most: gauge theory methods worked very efficiently in this setting. At the same time, their analogue in dimension three, which is known as the Floer homology, is yet to reveal its full potential. The Floer homology is in the focus of investigator's research efforts.