Seiberg-Witten invariants of three-manifolds

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Proposal DMS-0071820 Principal Investigator: Liviu I. Nicolaescu This project involves topological and geometric aspects of the theory of three-dimensional Seiberg-Witten monopoles and it addresses two general guiding questions. The first question is about the topological significance of the monopole count. Meng and Taubes have shown that on a three-manifold with nontrivial homology this count is given by Reidemeister-Turaev torsion. In the case of rational homology spheres numerical experiments with Seifert manifolds suggest that a suitably altered monopole count determine in a very elegant fashion both the Casson-Walker invariant and the Reidemeister-Turaev torsion. The first part of the project is devoted to proving this. The second question is about the geometric meaning of three-dimensional monopoles. The link of a complex surface singularity is endowed with a natural CR geometry and it is natural to ask how much of it does a monopole capture. More precisely we address the following fundamental question: how do monopoles "look like" as the link of the singularity collapses onto the exceptional divisor of a resolution of the singularity? The three-dimensional world has been a source of inspiration for many important developments in algebraic topology. It is within the reach of our physical intuition so some visualization is possible (think of a cube) yet there is plenty of room for surprises (think of gluing in pairs the opposite faces of the cube). One modern point of view is that, if presented in a favorable light, the three-dimensional world will reveal its mysteries. The mathematicians phrase this in a more sober way. If one can find a nice geometry on a three-manifold then one can unlock some of its topological features. The monopoles are geometric objects introduced to the mathematical world by physicists. One could think of them as "signals" produced by a three-dimensional space. These ``signals'' depend on how we look at the manifold. Since they appeared on the mathematical scene they have provided us with surprising insights about the structure of low-dimensional worlds. This project is about counting these "signals" in a meaningful way and then understanding the significance of each one of them individually when observed under a favorable light.