Partial fiber sum decompositions and signatures of Lefschetz fibrations

Abstract: In his Ph.D. thesis, Burak Ozbagci described an algorithm computing signatures of Lefschetz fibrations where the input is a factorization of the monodromy into a product of Dehn twists. In this note, we give a reformulation of Ozbagci's algorithm which becomes much easier to implement. Our main tool is Wall's non-additivity formula applied to what we call partial fiber sum decomposition of a Lefschetz fibration over 2-disk. We show that our algorithm works for bordered Lefschetz fibrations over disk and it yields a formula for the signature of branched covers where the branched loci are regular fibers. As an application, we give the explicit monodromy factorization of a Lefschetz fibration over disk whose total space has arbitrarily large positive signature for any positive fiber genus.