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A NOTE ON CONTACT STRUCTURES

We construct two contact structures on a homology sphere M , that are homotopic through 2-plane fields, but are not isotopic as contact structures. A diffeomorphism of M permutes these two contact structures. In [LM] Lisca and Matic gave examples of homotopic but not isotopic contact structures. In this note we give a simple construction of another such example. Unlike the examples of Lisca and Matic, in our example the group of diffeomorphism of the 3-manifold acts nontrivially on the set of path components of oriented contact structures of the manifold, while the action of this group is trivial on the set of path components of the oriented tangent plane distributions. Our construction of contact structures uses theorem of Y. Eliashberg, characterizing symplectic 4-manifolds with pseudo-convex boundary (PC -manifolds). Theorem 1 ([E]) : Let X = B ∪ (1− handles) ∪ (2− handles) be fourdimensional handlebody with one 0-handle and no 3or 4-handles. Then: • The standard PC structure on B can be extended over 1-handles so that manifold X1 = B 4 ∪ (1− handles) has pseudo-convex boundary. • If each 2-handle is attached to ∂X1 along a Legendrian knot with framing one less then Thurston-Bennequin framing of this knot, then the symplectic form on X1 can be extended over 2-handles to a symplectic form on X, which makes X a PC manifold. We can visualize pseudo-convex handlebodies as follows: Take a 0-handle to be the unit ball B in C with the induced symplectic structure; choose coordinates in R ⊂ S = ∂B, so that the induced contact structure ξ0 on R is the kernel of the form λ0 = dz+ xdy. We also assume that the centers of the attaching balls of each 1-handle lies on the plane {x = constant}. Projection of a Legendrian link on the plane {x = 0} has intersection only of the type shown on Figure 1 (coming from left handed crossing), no vertical tangencies and all minima and maxima in y-direction are cusps. Moreover, every projection with these properties is a projection of some Legendrian link. Thurston-Bennequin invariant of a Legendrian knot α can be calculated by: tb(α) = bb(α)− (number of right cusps) where bb(α) is the blackboard (yz-plane) framing of the projection of α. Any PC-manifoldX induces a contact structure on the boundary 3-manifold Y = ∂X by restricting the dual K∗ of the canonical line bundleK → X. Furthermore if α