Knots and Surfaces in four-manifolds

Introduction

Four-dimensional smooth manifolds show very different behavior 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.

Importance of the Genus Function

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:

1. Topological and symplectic/algebraic geometric methods (on the constructive side)
2. Algebraic and analytic methods resting on specific PDEs and on counting their solutions (on the obstructive side)

Research Objectives

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.

Candidate for an Invariant

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

Novel Approach

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.

Study of the Concordance Group

We plan to study divisibility and torsion questions in this group via knot Floer homology. The definition of the concordance group rests on the concept of slice knots, which is closely related to the ribbon construction.

Potential Counterexamples

We plan to further study potential counterexamples for the famous Slice-Ribbon conjecture. The proposed problems can also provide explanations of the special behavior of four-manifolds with definite intersection forms, like the four-sphere and the complex projective plane.