Knots and Surfaces in four-manifolds

Four-dimensional smooth manifolds show very different behaviour than manifolds in any other dimension.

In fact, in other dimensions we have a somewhat clear picture of the classification, while dimension four is still elusive. The project aims to further our knowledge in this question in several ways.

The genus function, and its enhanced version taking knots and their slice surfaces into account, plays a crucial role in understanding different smooth structures on four-manifolds.

Techniques for studying these objects range from topological and symplectic/algebraic geometric (on the constructive side) to algebraic and analytic methods resting on specific PDE’s and on counting their solutions (on the obstructive side).The proposal aims to study several interrelated questions in this area.

We plan to construct further exotic structures, detect and better understand their exoticness. In doing so, we put strong emphasis on knots and their slice properties in various four-manifolds.

Ultimately we provide a candidate for an invariant, which is a smooth (and somewhat complicated) generalization of the intersection form, and we expect this generalization to characterize smooth four-manifolds.

The novelty in this approach is the incorporation of knots and their slice surfaces in a significant and organized manner into the picture.

While it provides a refined tool in general, this approach also touches classical aspects of four-manifold topology through the study of the concordance group. We plan to study divisibility and torsion questions in this group via knot Floer homology.

Definition of the concordance group rests on the concept of slice knots, which is closely related to the ribbon construction. We plan to further study potential counterexamples for the famous Slice-Ribbon conjecture.

The proposed problems can also provide explanations of the special behaviour of four-manifolds with definite intersection forms, like the four-sphere and the complex projective plane.