Kernel of the variation operator and periodicity of open books

We consider a parallelizable 2n-manifold F which has the homotopy type of the wedge product of n-spheres and show that the group of pseudo-isotopy classes of orientation preserving diffeomorphisms that keep the boundary ∂F pointwise fixed and induce the trivial variation operator is a central extension of the group of all homotopy (2n+1)-spheres by Hn(F;Sπn(SO(n))). Then we apply this result to study the periodicity properties of branched cyclic covers of manifolds with simple open book decompositions and extend the previous results of Durfee, Kauffman and Stevens to dimensions 7 and 15.