# Synchronizing Smooth and Topological 4-Manifolds (16w5145)

Arriving in Banff, Alberta Sunday, February 21 and departing Friday February 26, 2016

## Organizers

(Michigan State University)

(Columbia University)

(Indiana University)

(Université du Québec à Montréal)

## Objectives

This workshop will bring together a diverse collection of researchers, with a particular emphasis on bridging a chasm between the knowledge bases of researchers studying 4-manifolds in the topological and smooth categories. Three overarching goals will guide our participant selection and workshop program:

1. Facilitate a fruitful interaction between researchers working on topological and smooth 4-manifolds.
2. Introduce participants to the diverse collection of techniques currently utilized in 4-dimensional topology, and facilitate a discussion about the scope and limitations of these techniques
3. Address fundamental problems for concordance and homology cobordism, and routes within established or conjectural frameworks for their attack.

To appreciate the timeliness of such a workshop, one must first be aware of a significant rift in the field of 4-manifolds arising from the equally deep, yet starkly different approaches taken in the study of topological and smooth 4-manifolds. Both approaches originated in the early 1980's, with the work of Freedman and Donaldson, respectively. The former draws on delicate infinite constructions used to analyze the disc embedding problem for topological 4-manifolds, while the latter has roots in geometric analysis and, in particular, in the study of moduli spaces of solutions to non-linear PDEs. Due to the depth of both approaches, researchers in the respective areas quickly diverged, and very little communication took place -- or was even possible -- until recently.

On the topological side, a special semester on 4-manifolds at the Max Planck Institute for Mathematics in Bonn in 2013 has brought about a resurgence in interest in the techniques of Freedman and Quinn. From this semester emerged a strong corps of young mathematicians trained in these techniques. On the smooth side, the remarkable developments of Heegaard Floer theory have led to a powerful collection of invariants for analyzing smooth concordance and homology cobordism which are simultaneously effective and accessible. A similarly talented group of researchers leads this area. Both camps are eager to share and utilize techniques, and initial inroads towards interaction of the techniques has already taken place (for instance, in the form of the bipolar filtration). We structure this workshop to serve as a catalyst for an effective interaction between smooth and topological techniques in 4-manifold theory, as viewed through the lens of concordance.

To facilitate this, the workshop will be front-loaded with expository talks given by leading experts from the smooth and topological camps, respectively, and by experts from singularity theory and the theory of Lefschetz fibrations and their generalizations. These talks will be aimed at giving an overview of the techniques available in these areas, and at introducing participants both to the key ideas and problems to facilitate a fertile mingling of ideas. Specific lectures will given on
• An overview of the disc embedding problem for 4-manifolds and its role in understanding sliceness in the topological category.
• Filtrations of concordance and cobordism groups coming from the derived series of the fundamental group.
• Broken fibrations of 4-manifolds and Cerf theory for Morse 2-functions.
• Heegaard Floer homology and the invariants it offers for concordance and homology cobordism.
• Khovanov homology, its generalizations, and their concordance content.
• Homology cobordism invariants derived from $Pin(2)$ equivariant gauge theory and their role in the solution to the triangulation conjecture

In addition to research lectures, we also hope to have informal and collaborative problem sessions to help steer participants towards developing strategies for the following types of problems.
• Determine the existence of torsion in concordance or cobordism groups of order other than two.
• Examine the validity of the ribbon slice conjecture, and the question of whether ribbonness of a knot can be algorithmically determined using three-dimensional methods.
• Determine means to sidestep the seemingly inherent chirality" of many of the invariants for smooth 4-manifold e.g. formulating a strategy for obstructing sliceness of the right-handed Whitehead double of the left-handed trefoil.
• Determine the extent to which the homology cobordism class of the complement of a knot determines its concordance class. For instance, decide if the homology cobordism classes of all Dehn surgeries on a knot or all of its branched cyclic covers, determines its concordance class (in either category).
• Investigate the smoothing theory of 4-manifolds with fundamental groups for which the topological classification is known.