Splitting singular fibers with periodic monodromies and their monodromy factorization

Abstract

A Lefschetz fibration is a smooth 4-manifold admitting a surface bundle structure over a surface except at finitely many singular fibers, whose singularities are only of nodal type. From the structure of the singular fibers, the monodromy of each singular fiber is given by a right-handed Dehn twist along a curve, called a vanishing cycle, in the fiber. Collecting all monodromy data from the singular fibers, we obtain a monodromy factorization into right-handed Dehn twists, which completely determines the Lefschetz fibration. In this paper, we construct a fibration with one singular fiber whose monodromy is periodic (that is, the monodromy homeomorphism is isotopic to a periodic map). Following the idea of Matsumoto, we give a splitting of the singular fiber into Lefschetz fibers and determine their vanishing cycles for a collection of periodic monodromies. We describe the construction of the splitting singular fibers and the procedure for reading vanishing cycles using two branched covering structures of the fibers. We also give splittings of singular fibers into Lefschetz fibers related to a family of periodic actions and determine their vanishing cycles.