A K-contact simply connected $\mathbf 5$-manifold with no semi-regular Sasakian structure

Abstract

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 $g>1$.