Topology of 3-cosymplectic manifolds

We continue the program of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds which are the closest odd-dimensional analogue of hyper-Kahler structures. We show that there is an action of the Lie algebra so(4,1) on the basic cohomology spaces of a compact 3-cosymplectic manifold with respect to the Reeb foliation. This implies some topological obstructions to the existence of such structures which is expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kahler factor and an abelian three dimensional Lie group. Nevertheless, we present a nontrivial example of compact 3-cosymplectic manifold which is not the global product of a hyper-Kahler manifold and a flat 3-torus.