This collaborative project will study the topology of smooth 4-dimensional manifolds, in connection with well-known problems in low-dimensional topology. We will focus on the construction of new smooth manifolds with symplectic structures, including Stein manifolds and symplectic fillings of certain contact 3-manifolds. Recent advances in techniques based on knot surgery and Luttinger surgery for creating exotic manifolds with small Euler characteristic will be coupled with computations of gauge-theoretic and symplectic invariants. We will make use of 4-dimensional handlebody techniques in these constructions, with an organizing principle being the search for 'corks' and 'plugs' as a technique for changing the smooth structure. Techniques of gauge theory and symplectic geometry will be used to investigate the classification of symplectic 4-manifolds and their symmetry groups.
The physical world of space and time is a 4-dimensional space whose local structure is well understood but whose large-scale (or topological) properties remain mysterious. This Focused Research Group will explore the global topology of 4-dimensional spaces, with a goal of understanding what kinds of spaces (called 4-dimensional manifolds) can exist as mathematical objects, and what the properties of such manifolds are. Of particular interest will be the problem of existence and uniqueness of symplectic structures, as well as that of determining the symmetries of a given manifold. The group will investigate how subtle changes in the smooth structure of a manifold can be achieved by gluing together pieces of different manifolds. Such changes will be detected by combining expertise from several disciplines, including powerful techniques derived from gauge theories of mathematical physics.
''. During the research period the PI mainly worked on the problem about how certain low dimensional results about contact structures, open books, Stein manifolds and Lefschetz fibrations can be generalized to higher dimensions. As a consequence, the PI obtained two results (explained below) both of which are joint work with his collaborator Selman Akbulut. In the first result, It was shown that if a contact open book $OB$ on a $(2n+1)$-manifold $M$ ($ngeq1$) is induced by a Lefschetz fibration $pi:W o D^2$, then there is a 1-1 correspondence between positive stabilizations of $OB$ and positive stabilizations of $pi$. Also any exact open book, an open book induced by a compatible exact Lefschetz fibration, carries a contact structure. Moreover, it was shown that there is a 1-1 correspondence (similar to the one above) between convex stabilizations of an exact open book and convex stabilizations of the corresponding compatible exact Lefschetz fibration. We also show that convex stabilization of compatible exact Lefschetz fibrations produces symplectomorphic completions. In the second result, it was shown that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on $W$, any simply connected embedded Lagrangian submanifold of a page in a convex open book on $partial W$ can be assumed to be Legendrian in $partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on $W$ yields a compact convex Lefschetz fibration on a Liouville domain $W'$ which is exact symplectomorphic to a positive expansion of $W$. In particular, with the induced structures $partial W$ and $partial W'$ are contactomorphic. This results, in particular, proves that there is a well-defined map from the category of Lefschetz fibrations into the category of Stein domains. PI: Mehmet Firat Arikan.
