Tight contact structures and Seiberg-Witten invariants

摘要

Contact structures are the odd-dimensional analogue of symplectic structures. Although much is known, the present understanding of both kinds of structures is far from complete, even in low dimensions. Due to the work of Cliff Taubes is it now a fact that there is a close relationship between symplectic structures and Seiberg-Witten monopole equations on 4-manifolds [Ta1, Ta2, Ta3, Ta4]. The results contained in this paper may be thought of as evidence that the 3-dimensional reduction of the SeibergWitten equations is related with contact structures. Although we will not work directly with the 3-dimensional version of the equations, we will use the 4-dimensional Seiberg-Witten theory to prove new results about contact structures on 3-manifolds.