Structures de contact sur les varietes fibrees en cercles au-dessus d'une surface

Abstract

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isotopic to a contact structure transverse to the fibers or there exists, in some finite sheeted cover of V, a Legendrian curve isotopic to the fiber along which $ξ$ determines the same framing as the fibration $V \to S$. In Section 3, we classify contact structures that are transverse to the fibers up to isotopy and conjugation. In Section 4, we study general tight contact structures on V. We prove that virtually over-twisted contact structures form finitely many isotoy classes while isotopy classes of universally tight contact structures are in one-to-one correspondence with isotopy classes of systems of essential curves on S.