On the construction of K-contact non-Sasakian Smale-Barden manifolds

Abstract

In the breakthrough paper \cite{Mu-jems}, it is constructed the first example of a simply connected compact $5$-manifold (aka.\ Smale-Barden manifold) which admits a K-contact structure but does not carry a Sasakian structure. In this work we clarify some aspects of the construction of \cite{Mu-jems}, determining explicitly the number $N$ of symplectic surfaces needed to have an isotropy locus that produce a $5$-manifold that is K-contact but not Sasakian. Also, in order to analyse the geography problem of determining which Smale-Barden manifolds admit K-contact but not Sasakian structures, we refine and generalize the constructions of symplectic surfaces in a symplectic $4$-manifold with transversal intersections giving rise to such manifolds.