Periodicity of branched cyclic covers of manifolds with open book decomposition

Abstract

In this paper we study branched cyclic covers of manifolds with open book decomposition. Such a decomposition splits a manifold M in a binding K and a complement which fibres over S ~ with as fibre the page F such that the fibration is trivial in a neighbourhood of K. For example, if M is a sphere, we get a fibered knot K. Open book decompositions also arise in singularity theory: i fX is a smoothing of the germ of an isolated complex singularity, i.e. a one-parameter deformation for which the general fibre is smooth, then the link M of the singular point in X has an open book decomposition with binding the link of the smoothed germ, while the fibration is the same as the Milnor fibration. The cyclic k-fold cover of M, branched along K, is induced from the k-fold cover of S 1 over S 1. We study these k-fold covers in terms of the monodromy h : F~F of the fibration and the topology of K. There is a close connection between the topology of M and F if M has odd dimension 2n + 1 and F has the homotopy type of a n-dimensional CW-complex; the homology of F is then determined by M and vice versa except in the middle dimension, where we have the variation exact sequence: