In , the first and the third authors introduced new classes in the Johnson cokernels of the mapping class groups of surfaces by a representation theoretic approach based on some previous results for the Johnson cokernels of the automorphism groups of free groups. On the other hand, in , Kawazumi and the second author introduced another type of classes by a topological consideration of self-intersections of curves on a surface. In this paper, we show that the classes found in  are contained in the classes found in  in a stable range. Furthermore, we prove that the anti-Morita obstructions [14m+1] for m≥1 obtained in  and a hook-type component [3,15] detected in  appear in their gap.
- Johnson homomorphism
- Mapping class groups