  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
$\partial F$ pointwise fixed and induce the trivial variation operator is a
central extension of the group of all homotopy $(2n+1)$-spheres by $H_n\bigl(F;
S\pi_n(SO(n))\bigr)$. 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.
