Convex Decomposition Theory 3

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures. It has been known for some time that there are deep connections between the theory of taut foliations and tight contact structures due to the work of Eliashberg and Thurston [7]. In particular, they proved that a taut foliation can be perturbed into a (universally) tight contact structure. In a previous paper [19] we introduced the notion of convex decompositions and explained how convex decompositions can naturally be viewed as generalizations of sutured manifold decompositions introduced by Gabai in [8]. In this paper we take the viewpoint that convex decompositions are completely natural in 3-dimensional contact topology and that many theorems can be proven directly in the category of tight contact manifolds with convex splittings as morphisms. The first theorem of the paper is a version of a theorem by Gabai-Eliashberg-Thurston in the case of manifold with boundary: Theorem 0.1 (Gabai-Eliashberg-Thurston). Let (M, γ) be an oriented, compact, connected, irreducible, sutured 3-manifold which has boundary, is taut, and has annular sutures. Then (M, γ) carries a universally tight contact structure. We provide an alternate proof which (1) does not require us to perturb taut foliations into tight contact structures as we did in [19] and (2) does not resort to four-dimensional symplectic filling techniques in order to prove tightness. Instead, we use Gabai’s sutured manifold decomposition and directly apply a gluing theorem (Theorem 1.6). To prove the gluing theorem we apply key ideas from Colin’s papers [3] and [4] in the context of convex decomposition theory. We also apply similar ideas to prove the following theorem: Theorem 0.2. Let M be an oriented, closed, connected, irreducible 3-manifold which contains an incompressible torus. Then M carries infinitely many isomorphism classes of universally tight contact structures. This theorem confirms a conjecture which has its beginnings in the works of Giroux [10] and Kanda [23], was proved for torus bundles over S by Giroux [11], and was extended Date: This version: January 31, 2001. 1991 Mathematics Subject Classification. Primary 57M50; Secondary 53C15.