Higher dimensional contact topology via holomorphic disks

TL;DR

This paper explores how holomorphic curves can be used in higher-dimensional contact topology to determine whether certain contact manifolds admit symplectic fillings, providing tools to identify non-fillable cases.

Contribution

It introduces methods using holomorphic disks to analyze symplectic fillings in higher-dimensional contact manifolds, advancing understanding in contact topology.

Findings

Certain contact manifolds are proven not to admit symplectic fillings.

Holomorphic curves serve as effective tools in higher-dimensional contact topology.

The approach extends techniques from low-dimensional cases to higher dimensions.

Abstract

These notes are based on a course that took place at the Universit\'e de Nantes in June 2011 during the "Trimester on Contact and Symplectic Topology". We will explain how holomorphic curves can be used to study symplectic fillings of a given contact manifold. Our main goal consists in showing that certain contact manifolds do not admit any symplectic filling at all.