Abstracts for Workshop on 4-manifolds, September 16 - 20, 2019

The smooth symplectic isotopy problem asks for the classification of smooth symplectic surfaces of the complex projective plane up to isotopy. This problem remains open in degrees greater than 17. Here we will consider a singular version of the problem--classifying symplectic submanifolds of $\mathbb{CP}^2$ with singularities of specified types given by cones on torus knots. We will show that for certain rational cuspidal curves, the classification can be made completely. This is based on joint work with Marco Golla.


 

We use the branched cover construction and ideas from connected Floer homology to define concordance invariants of knot in $S^3$. Calculations can be performed for double branched covers, in which case the invariants are trivial for alternating and torus knots and non-trivial for some pretzel knots. This allows us to derive some independence results in the smooth concordance group of knots.
 

Throughout this talk, we will compare the notions of topological isotopy, smooth isotopy, and smooth equivalence (via an ambient diffeomorphism preserving homology) between homotopic $2$-spheres smoothly embedded in a $4$-manifold. In particular, we will construct pairs of spheres that are smoothly equivalent but not even topologically isotopic. Indeed, our examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.
 

In this talk, I will explain a method to construct families of diffeomorphisms of a 4-manifold by using a 4D analogue of Goussarov-Habiro's theory of graph surgery in 3D. Our graph surgery would produce lots of potentially nontrivial elements of homotopy groups of diffeomorphism groups of 4-manifolds. I will discuss about their applications to 4D analogue of the Smale conjecture and the 4D light bulb theorem for 3-disk.