Exact Lagrangian Fillings of Legendrian links and Weinstein 4-manifolds

Abstract

One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. In this thesis we prove that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings which are smoothly isotopic but not smoothly isotopic. To distinguish these fillings we use Floer theoretic techniques developed by Casals and Ng. We provide one of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. As part of a collaboration, we also explore obstructions to the existence of exact Lagrangian cobordisms between Legendrian links that can be applied to obstructing certain immersed exact Lagrangian fillings.Weinstein domains are examples of a symplectic manifold with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein $4$-dimensional domains can be represented with Weinstein handlebody diagrams of Legendrian links in $(\#^m(S^1\times S^2), \xi_{std})$ or $(S^3, \xi_{std}).$ Studying the symplectic topology of Weinstein domains has allowed for new perspectives when studying various manifolds including complex affine varieties. We study the Milnor fibers $M_f$ of isolated unimodular singularities. Keating constructed an exact Lagrangian torus in $M_f$. We show that there are exact infinitely many Hamiltonian non-isotopic Lagrangian tori in $M_f$ using Weinstein handlebody diagrams and exact Lagrangian fillings of Legendrian links. We also show that $M_f$ contains a new infinite set of symplectically knotted Lagrangian spheres. Additionally, we provide a generalization of a criterion for when the symplectic homology of a Weinstein $4$ manifold is non-vanishing given a Weinstein handlebody diagram. Finally, we provide a summary of a second collaboration which studies the complement of smoothings of toric divisors in toric $4$-manifolds. We show that for certain smoothings, these complements have a particular Weinstein structure, and we provide an algorithm to construct the Weinstein handlebody diagram of such complements.