Equivariant Gauge Theory on 3-Manifolds

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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.