Abstract

Contact topology in dimension three is shaped by the fundamental dichotomy between “tight” and “overtwisted” contact structures, and while it is not known whether any such dichotomy exists in higher dimensions, there are certainly contact structures in all dimensions that have all the trappings of overtwistedness (e.g. nonfillability, vanishing contact homology), or tightness (e.g. admitting a Reeb vector field with no contractible orbits). In dimension 3, the invariant known as “Giroux torsion” has played a central role in classifying tight contact structures, and in this talk I will explain how one can generalize it to find the first examples in all dimensions of contact structures that must be considered tight but do not admit any symplectic fillings. A crucial ingredient for this is the existence (also in all dimensions) of symplectic manifolds with disconnected convex boundary, which requires a surprising digression into algebraic number theory. This is joint work with Patrick Massot and Klaus Niederkrueger.