On the cohomology rings of holomorphically fillable manifolds

An odd-dimensional differentiable manifold is called \emph{holomorphically fillable} if it is diffeomorphic to the boundary of a compact strongly pseudoconvex complex manifold, \emph{Stein fillable} if this last manifold may be chosen to be Stein and \emph{Milnor fillable} if it is diffeomorphic to the abstract boundary of an isolated singularity of normal complex analytic space. We show that the homotopical dimension of a manifold-with-boundary of dimension at least 4 restricts the cohomology ring (with any coefficients) of its boundary. This gives restrictions on the cohomology rings of Stein fillable manifolds, on the dimension of the exceptional locus of any resolution of a given isolated singularity, and on the topology of smoothable singularities. We give also new proofs of structure theorems of Durfee & Hain and Bungart about the cohomology rings of Milnor fillable and respectively holomorphically fillable manifolds. The various structure theorems presented in this paper imply that in dimension at least 5, the classes of Stein fillable, Milnor fillable and holomorphically fillable manifolds are pairwise different.