On the Coarse Classification of Tight Contact Structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures. In this article we explain how to normalize tight contact structures with respect to a fixed triangulation. Using this technique, we obtain the following results: Theorem 0.1. Let M be a closed, oriented 3-manifold. There are finitely many homotopy classes of 2-plane fields which carry tight contact structures. Theorem 0.2. Every closed, oriented, atoroidal 3-manifold carries a finite number of tight contact structures up to isotopy. P. Kronheimer and T. Mrowka [KM] had previously shown Theorem 0.1 for (weakly) symplectically (semi-)fillable contact structures. Our theorem is a genuine improvement of the Kronheimer-Mrowka theorem because there exist tight structures which are not fillable [EH]. Now, since every Reebless foliation is a limit of tight contact structures [Co4, ET], we obtain a new proof of a recent result of D. Gabai [Ga]. Corollary 0.3 (Gabai). There are finitely many homotopy classes of plane fields which carry Reebless foliations. Next, shifting our attention to isotopy classes of contact structures, we see that Theorem 0.2 complements the following theorem [Co2, Co3, HKM]: Theorem 0.4 (Colin, Honda-Kazez-Matić). Every closed, oriented, irreducible, toroidal 3manifold carries infinitely many tight contact structures up to isomorphism. Summarizing, we have: Theorem 0.5. A closed, oriented, irreducible 3-manifold carries infinitely many tight contact structures (up to isotopy or up to isomorphism) if and only if it is toroidal. A more complete account of the proofs will appear in [CGH]. Acknowledgements. This project was started in the fall of 2000 when the authors visited the American Institute of Mathematics and Stanford University during the Workshop on Date: This version: May 30, 2002. 1991 Mathematics Subject Classification. Primary 53D35; Secondary 53C15.