Abstract
Abstract Kollár has found subtle obstructions to the existence of Sasakian structures on five-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 $H_1(M,\mathbb{Z})=0,$ which is K-contact but which carries no semi-regular Sasakian structures.
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