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.