Topology of 4-manifolds, links and Engel groups

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Topology of manifolds is a subject aimed at classification of shapes (manifolds) which are locally modeled on the Euclidean space of a given dimension. The topology of 4-dimensional manifolds is of particular interest since this dimension is relevant in physics while studying space-time, and it is the subject of some of the outstanding open problems in Topology. This dimension is also special in the sense that it is borderline between "low dimensions" where many classification results have been settled and "higher dimensions" where geometric classification problems are known to admit an algebraic reformulation. This project is aimed at classification up to continuous deformations of 4-manifolds with special characteristics, known as large fundamental groups. The main part of this project concerns geometric classification of topological 4-manifolds, specifically the topological surgery conjecture and the s-cobordism conjecture for "large" fundamental groups. The notion of a 1/2-null surgery problem is proposed, as an intermediate step in the context of universal surgery problems. A key ingredient is the group-theoretic Engel relation, which is closely related to higher-order singularities of surfaces in 4-manifolds. The framework of the AB-slice problem is also considered. Another part of the project concerns ongoing work on applications of (2+1)-dimensional topological quantum field theories to combinatorics and to representations of mapping class groups. The PI also plans to consider questions in geometric and topological complexity of embeddings in dimension 4.