Meyer's signature cocycle and hyperelliptic fibrations

Abstract

We show that the cohomology class represented by Meyer's signature cocycle is of order \(2g+1\) in the 2-dimensional cohomology group of the hyperelliptic mapping class group of genus \(g\). By using the \(1\)-cochain cobounding the signature cocycle, we extend the local signature for singular fibers of genus 2 fibrations due to Y. Matsumoto [18] to that for singular fibers of hyperelliptic fibrations of arbitrary genus \(g\) and calculate its values on Lefschetz singular fibers. Finally, we compare our local signature with another local signature which arises from algebraic geometry.