  Koll\'ar has found subtle obstructions to the existence of Sasakian
structures on 5-dimensional manifolds. In the present article we develop
methods of using these obstructions to distinguish K-contact manifolds from
Sasakian ones. In particular, we find the first example of a closed 5-manifold
M with $H_1(M,Z)=0$ which is K-contact but which carries no semi-regular
Sasakian structures.
