Berkeley topology seminar

Little is known about tight contact structures which are not fillable.  Algebraic torsion measurements in embedded contact homology are useful for obstructing symplectic fillability and overtwistedness of the contact 3-manifold, but has been left unexplored. We discuss the methods we developed, focusing on concave linear plumbings of symplectic disk bundles over spheres admitting a concave contact boundary. This talk is based on joint work with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  Time permitting, we will discuss our work in progress to determine nonfillable tight contact 3-manifolds obtained from more general plumbings.

A pseudo-isotopy is a weakening of the concept of isotopy, removing the insistence that it is level-preserving. Pseudo-isotopies play a key role in the study of mapping class groups in dimensions 4 and above. We investigate the question: if a self-diffeomorphism of a 4-manifold is topologically pseudo-isotopic to the identity, must it always be smoothly so? We produce the first examples where the answer is “no” -- the first exotic pseudo mapping classes. On the other hand, we derive a set of conditions on the fundamental group of a 4-manifold that identify a large class of examples where this topological to smooth upgrade can always be made. I will explain how these results make progress on our original motivating question of whether topological isotopy implies smooth stable isotopy for diffeomorphisms of 4-manifolds. This is joint work with Mark Powell and Oscar Randal-Williams.

An aspherical space is one with vanishing higher homotopy groups; under mild assumptions this means that it is determined up to homotopy type by its fundamental group. A famous conjecture of Borel states that closed aspherical manifolds are in fact determined up to homeomorphism by their fundamental group. One could also ask (although Borel famously didn’t) whether smooth aspherical manifolds are determined up to diffeomorphism by their fundamental group; this is known to hold in dimensions at most 3 and to be false in dimension at least 5. We resolve the remaining case by exhibiting closed smooth aspherical 4-manifolds that are homeomorphic but not diffeomorphic. This is joint with Davis, Hayden, Huang, and Sunukjian.