This research project concerns the geometric classification theory of topological 4-manifolds. The classification techniques are known to hold for a class of good fundamental groups, and are conjectured to fail in general. This conjecture remains a central open 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 been a 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, and at proving the non-existence result of certain 4-dimensional shapes with large fundamental groups, which contain many loops that cannot be contracted.
