On a function on the mapping class group of a surface of genus 2

We study representations of subgroups of the mapping class group Mg of a surface of genus g≥2 arising from the actions of them on the first cohomology groups of the surface with local coefficient systems which are defined by nontrivial homomorphisms π1(Σg,∗)→Z2=Aut(Z). As an application, in the case of g=2, we construct a function on M2 which agrees with the Meyer function φ:M2→Q on the Torelli group J2.