Tight Contact Structures via Dynamics

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) ows 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 de nitions and theorems related to the geometry and dynamics of contact structures see, e.g., [1]. A contact structure on a 3-manifoldM is a totally non-integrable 2-plane eld in TM . More speci cally, at each point p 2 M we have a 2-plane p TpM that varies smoothly with p, with the property that is nowhere integrable in the sense of Frobenius: i.e., there exists (locally) a de ning 1-form (whose kernel is ) such that ^ d 6= 0. If is globally de ned, is called orientable and a contact 1-form for . We adopt the common restriction to orientable contact structures. The interesting (and di cult) 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 di eomorphic preserving the plane elds. A similar result holds for a surface in a contact manifold (M; ) as follows. Generically, Tp \ p will be a line in Tp : This line eld integrates to a singular foliation called the characteristic foliation of . One can show, as in the single-point case of Darboux's Theorem, that determines the germ of along . There has recently emerged a fundamental dichotomy in three dimensional contact geometry. A contact structure is overtwisted if there exists an embedded disk D in M whose characteristic foliation D contains a limit cycle. If is not overtwisted then it is called tight. Eliashberg [6] has completely classi ed overtwisted contact structures on closed 3-manifolds | the geometry of overtwisted contact structures reduces to the algebra of homotopy classes of plane elds. Such insight into tight contact structures is slow in coming. The only general method for constructing tight structures is by Stein llings (see [14, 7]) and the uniqueness question has only been answered on S [8], T 3 [13, 20], most T -bundles over S [13], and certain lens spaces L(p; q) [10]. Thus we have the fundamental open question: does every 3-manifold M admit a tight contact structure? Martinet [22] and Thurston and Winkelnkemper [26] have used surgery techniques to show that all closed 3-manifolds admit contact 1991 Mathematics Subject Classi cation. Primary: 53C15, 57M12; Secondary: 58F05.