Tight contact structures via dynamics

Abstract

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched covering, may be performed on dynamically-constrained links in such a way as to preserve a transverse tight contact structure. 1. CONTACT GEOMETRY AND DYNAMICS For a more thorough treatment of the basic definitions and theorems related to the geometry and dynamics of contact structures see, e.g., [1]. A contact structure 5 on a 3-manifold M is a totally nonintegrable 2-plane field in TM. More specifically, at each point p E M we have a 2-plane ~p C TpM that varies smoothly with p, with the property that 5 is nowhere integrable in the sense of Frobenius, i.e., there exists (locally) a defining 1-form ac (whose kernel is J) such that a A doa 7 0. If a is globally defined, ~ is called orientable and a a contact 1-form for ~. We adopt the common restriction to orientable contact structures. The interesting (and difficult) problems in contact geometry are all of a global nature: Darboux's Theorem (see, e.g., [23, 1]) implies that all contact structures are locally contactomorphic, or diffeomorphic preserving the plane fields. A similar result holds for a surface E in a contact manifold (M, ) as follows. Generically, TpE n $p will be a line in TpE. This line field integrates to a singular foliation SE called the characteristic foliation of S. One can show, as in the single-point case of Darboux's Theorem, that >E determines the germ of . along S. There has recently emerged a fundamental dichotomy in three dimensional contact geometry. A contact structure S is overtwisted if there exists an embedded disk D in M whose characteristic foliation D1 contains a limit cycle. If ~ is not overtwisted, then it is called tight. Eliashberg [6] has completely classified overtwisted contact structures on closed 3-manifolds the geometry of overtwisted contact structures reduces to the algebra of homotopy classes of plane fields. Such insight into tight contact structures is slow in coming. The only general method for constructing tight structures is by Stein fillings (see [14, 7]), and the uniqueness Received by the editors January 28, 1998. 1991 Mathematics Subject Classification. Primary 53C15, 57M12; Secondary 58F05.