Equivariant Gauge Theory on 3-Manifolds

使用Floer同调研究三维流形的主要障碍是缺乏对三维拓扑标准结构（如Dehn手术、连通和等）的行为的清晰理解。在这个方向上已知的结果在花同源性计算或应用中没有多大进展。研究者的主要论点是，相对于花同源性行为最自然的操作是分支覆盖。因此，研究者研究了具有群作用的3-流形的规范理论，并将规范理论不变量与分支集的经典不变量联系起来。例如，他先前对拟齐次曲面奇点连杆的等变规范理论的研究，得到了一个用经典不变量表示其flower同调的封闭形式公式。研究了具有循环群作用的整同调球的等变卡森不变量，并将其与经典卡森不变量和结特征联系起来。该定义利用了等变规范理论，特别是等变弗洛尔指数。在反全纯对合的情况下，后者与Stein曲面的拓扑和非紧化Kaehler几何密切相关。研究者还继续他的研究完全相交奇点的链接，以及等变卡森不变量与同调协的相关性，以及五维及更高维拓扑流形的三角化猜想。本研究是利用规范理论的方法对三维和四维流形的研究，来源于理论物理。在过去的二十年里，这些方法给经典的低维拓扑带来了新的生命，导致了许多引人注目的发展，并解决了许多难题。这一领域发生了革命性的变化，以至于20年前的知识状态现在在许多方面看起来就像一片几乎完全无知的沙漠。四维流形受益最大：规范理论方法在这种情况下非常有效。与此同时，它们在三维空间的类似物，也就是众所周知的弗洛尔同源性，还没有显示出它的全部潜力。弗洛尔同源性是研究者研究的重点。

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.