  We construct the first example of a 5-dimensional simply connected compact
manifold that admits a K-contact structure but does not admit a 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 but one of genus 1 and the other 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 when all of them but one are of genus 1 and the other of genus g>1.
