Topology of 3-cosymplectic manifolds

We continue the programme of Chinea, De León and Marrero who studied the topology of cosymplectic manifolds.We study 3-cosymplectic manifolds that are the closest odd-dimensional analogue of hyper-Kähler 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 are expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kähler factor and an abelian three-dimensional Lie group. Nevertheless, we present a non-trivial example of a compact 3-cosymplectic manifold that is not the global product of a hyper-Kähler manifold and a flat 3-torus.