Classification Theory of 4-Manifolds

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This research project concerns the geometric classification theory of topological 4-manifolds. The classification techniquesare known to hold for a class of good fundamental groups, and are conjectured to fail in general. This conjecture remains a centralopen problem in 4-dimensional topology.One goal of this project is to prove that the disk embedding theorem, and consequently the surgery and s-cobordism theorems, hold for amenable fundamental groups. Another approach to surgery, explored in the project, is provided by the disk embedding conjecture up to s-cobordism. The topology of 4-manifolds is closely related to link-slicing problems, and the third part of the project aims at extending the results about slicing to Whitehead doubles of homotopically trivial links. The classification of possible large-scale structures in dimension 4, which locally look like the 4-dimensional space-time, has beena focus of intensive research over the past 20 years. This area of research lies at the intersection of topology, geometry, analysis and physics. This project is aimed at classification of 4-dimensional objects with amenable fundamental groups, andat proving the non-existence result of certain 4-dimensional shapes with large fundamental groups, which contain many loops that cannot be contracted.