Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions,

with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find

obstructions for a closed manifold to admit such types of structures and in particular, to construct

K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the

hardest dimension is 5, where Kollar has found subtle obstructions to the existence of Sasakian

structures, associated to the theory of algebraic surfaces.

In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in

dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number 0 which is K-contact but which carries no semi-regular Sasakian structure.

(Joint work with J.A. Rojo and A. Tralle).