Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Abstract

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space CP 2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

Homology Smale–Barden Manifolds with K-contact but not Sasakian Structures

“…Remark 22. Theorem 21 corrects a statement of [17], where it is claimed that a K-contact structure can be constructed from an orbifold where the isotropy locus is not a symplectic surface. This is wrongly used to construct examples of K-contact manifold, and to conclude that the manifolds of [18] admit K-contact structures, which is the main result of [17].…”

“…In [17] it is claimed a solution of this question, but unfortunately the main result of that paper is flawed, as we explain in Remark 22. In the present paper we make the first step towards a positive answer for the above question. A homology Smale-Barden manifold is a compact 5-dimensional manifold with H 1 (M, Z) = 0.…”

“…In [17] it is claimed a solution of this question, but unfortunately the main result of that paper is flawed, as we explain in Remark 22.…”