Applications of Gauge Theory and Floer Homology to Low-Dimensional Topology

Brandeis University, Waltham MA

Investigators

Abstract

The understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Many of the questions posed by geometers and topologists have to do with the nature of three-dimensional spaces sitting in a four-dimensional space, and with the restrictions that the global properties of larger space places on such sub-spaces. The research in this National Science Foundation funded project uses modern tools of analysis and geometry to shed light on the local nature of such subspaces, including new methods for showing that singularities in such spaces cannot be smoothed. Related analytical techniques will be used to explore the global topology of four-dimensional spaces, including an investigation of their symmetries. Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological techniques. The first parts of the project, joint with Jianfeng Lin and Nikolai Saveliev, are concerned with the smooth topology of four-manifolds that homologically resemble a product of a three-dimensional manifold with a circle. Central questions concern the interpretation of the classical Rohlin invariant and multi-signature invariants in terms of gauge theory; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine on embeddings of punctured three-manifolds in four-space, using refined techniques from Heegaard Floer theory to find obstructions. An ongoing project with David Auckly, Hee Jung Kim, and Paul Melvin is concerned with the topology of the diffeomorphism group of a four-dimensional manifold and how it is affected by stabilization of the manifold. Finally, the PI will work with Saveliev and Demetre Kazaras on a new technique to obstruct the existence of positive scalar curvature cobordisms between even-dimensional manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.