Milnor fillable contact structures are universally tight

Abstract

We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tig ht contact structures that are not deforma- tions of taut (or Reebless) foliations. This answers two questions of Etnyre in (12). Let (X, x) be a normal complex surface singularity. Fix a local embedding of (X, x) in (C N , 0). Then a small sphere S 2N−1 e ⊂ C N centered at the origin intersects X transversely, and the complex hyperplane distribution ξcan on M = X ∩ S 2N−1 e induced by the complex structure on X is called the canonical contact structure. For sufficiently small radius e, the contact manifold is independent of e and the embedding, up to isomorphism. The 3-manifold M is called the link of the singularity, and (M, ξcan) is called the contact boundaryof (X, x). A contact manifold (Y, ξ) is said to be Milnor fillable if it is isomorphic to the contact bound- ary (M, ξcan) of some isolated complex surface singularity (X, x). In addition, we say that a closed and oriented 3-manifold Y is Milnor fillable if it carries a contact structure ξ so that (Y, ξ) is Milnor fillable. It is known that a closed and oriented 3-ma nifold is Milnor fillable if and only if it can be obtained by plumbing according to a weighted graph with negative definite intersection matrix (cf. (25) and (18)). Moreover a ny 3-manifold has at most one Milnor fillable contact structure up to isomorphism(cf. (5)). Note that Milnor fillable contact structures are Stein fillable (see (4)) and hence tight (10). Here we prove that every Milnor fillable contact structure is in fact universally tight, i.e ., the pullback to the universal cover is tight. We would like to point out that universal tightness of a contact structure is not implied by any other type of fillability. In (12), Etnyre settled a question of Eliashberg and Thurston (11) by proving that every contact structure on a closed oriented 3-manifold is obtained by a deformation of a foliation and raised two other related questions: