Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

Abstract

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus 3 , 4 3, 4 and 5 5 which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.