For the handlebody Torelli groups HI and HBI, the kernels of the Johnson-detected cup product maps in second rational cohomology are computed explicitly as sums of irreducible SL_g(Q) modules.
The second Johnson homomorphism and the second rational cohomology of the Johnson kernel
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abstract
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.
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Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
For the handlebody Torelli groups HI and HBI, the kernels of the Johnson-detected cup product maps in second rational cohomology are computed explicitly as sums of irreducible SL_g(Q) modules.