An Sp(2g;Z2)-module structure of the cokernel of the second Johnson homomorphism

The Johnson homomorphisms τk (k⩾1) give Abelian quotients of a series of certain subgroups of the mapping class group of a surface. Morita determined the rational image of the second Johnson homomorphism τ2. In this paper, we study the structure of the torsion part of the cokernel of τ2. First, we determine the rank of the cokernel over Z2. Although we do it first by computing explicitly, later we improve the proof, using the Birman–Craggs homomorphism, obtained by the classical Rohlin invariant of homology 3-spheres. Since τ2 is equivariant with respect to the action of the mapping class group, Im τ2 is Sp(2g;Z)-invariant and hence Sp(2g;Z) acts on the cokernel. Moreover, computing this action explicitly, we show that the action reduces to that of the finite symplectic group Sp(2g;Z2).