Columbia Symplectic geometry, Gauge theory, and Topology Seminar

We give a sketch of a proof that link concordance implies link homotopy. We explain the consequence and then dive into the construction of immersed Morse theory. This is a joint result with Mark Powell and Peter Teichner.

The rational blowdown operation in 4-manifold topology replaces a neighborhood of a configuration of spheres by a rational homology ball. This is a symplectic surgery that is often used to create exotic 4-manifolds. Configurations of spheres that can be rationally blown down typically arise from resolutions of surface singularities that admit rational homology disk smoothings. It is an interesting question to describe all such singularities. Stipsicz-Szabo-Wahl and Bhupal-Stipsicz studied QHD smoothings by topological and symplectic methods that suggest that QHD smoothings and Stein fillings for singularity links behave similarly. However, we find examples of unexpected rational blowdowns, where this similarity between smoothings and Stein fillings breaks down. Inspiration for our constructions comes from de Jong-van Straten's work on sandwiched singularities and its symplectic analog. The talk will focus on topology and include the necessary basics on singularities. (Joint with M. Beke and L. Starkston.)

Donaldson and Freedman's breakthroughs in the 1980s showed the existence of exotic pairs of orientable 4-manifolds. Since then, the study of exotica has been a major theme. Most methods for distinguishing exotic pairs have come from gauge theory or Floer theory, and combinatorial methods have been sought. In this talk, I will first discuss previous work on using Khovanov homology to distinguish exotica combinatorially, including work of Rasmussen, Hayden–Sundberg, and Ren–Willis. Then, I will present two new and simple combinatorial methods for distinguishing exotic pairs of compact, orientable 4-manifolds.