  The Johnson kernel is the subgroup of the mapping class group of a surface
generated by Dehn twists along bounding simple closed curves, and has the
second Johnson homomorphism as a free abelian quotient. In terms of the
representation theory of the symplectic group, we give a complete description
of cup products of two classes in the first rational cohomology of the Johnson
kernel obtained by the rational dual of the second Johnson homomorphism.
