A K-CONTACT SIMPLY CONNECTED 5-MANIFOLD WITH NO SEMI-REGULAR SASAKIAN STRUCTURE

We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit any semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them of genus 1 except for one of genus g > 1; (b) to prove a bound on the second Betti number b(2) of an algebraic surface with b(1) = 0 and having disjoint complex curves spanning the homology, all of them of genus 1 except for one of genus g > 1.