Seiberg-Witten invariants of three-manifolds

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Proposal DMS-0071820 Principal Investigator: Liviu I. NicolaescuThis project involves topological and geometric aspects of thetheory of three-dimensional Seiberg-Witten monopoles and itaddresses two general guiding questions. The first question isabout the topological significance of the monopole count. Mengand Taubes have shown that on a three-manifold with nontrivialhomology this count is given by Reidemeister-Turaev torsion. Inthe case of rational homology spheres numerical experiments withSeifert manifolds suggest that a suitably altered monopole count determine in a very elegant fashion boththe Casson-Walker invariant and the Reidemeister-Turaev torsion.The first part of the project is devoted to proving this. Thesecond question is about the geometric meaning ofthree-dimensional monopoles. The link of a complex surfacesingularity is endowed with a natural CR geometry and it isnatural to ask how much of it does a monopole capture. Moreprecisely we address the following fundamental question: how domonopoles "look like" as the link of the singularity collapsesonto the exceptional divisor of a resolution of the singularity?The three-dimensional world has been a source of inspirationfor many important developments in algebraic topology. It iswithin the reach of our physical intuition so some visualizationis possible (think of a cube) yet there is plenty of room forsurprises (think of gluing in pairs the opposite faces of thecube). One modern point of view is that, if presented in afavorable light, the three-dimensional world will reveal itsmysteries. The mathematicians phrase this in a more sober way.If one can find a nice geometry on a three-manifold then one canunlock some of its topological features. The monopoles aregeometric objects introduced to the mathematical world byphysicists. One could think of them as "signals" produced by athree-dimensional space. These ``signals'' depend on how we lookat the manifold. Since they appeared on the mathematical scenethey have provided us with surprising insights about the structureof low-dimensional worlds. This project is about counting these"signals" in a meaningful way and then understanding thesignificance of each one of them individually when observed undera favorable light.