A remark about weak fillings

Abstract

Let L be a closed manifold of dimension n≥2 which admits a totally real embedding into Cn. Let ST*L be the space of rays of the cotangent bundle T*L of L, and let DT*L be the unit disk bundle of T*L defined by any Riemannian metric on L. We observe that ST*L endowed with its standard contact structure admits weak symplectic fillings W which are diffeomorphic to DT*L and for which any closed Lagrangian submanifold N⊂W has the property that the map H1(N,R)→H1(W,R) has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.