A comparison of classes in the Johnson cokernels of the mapping class groups of surfaces

In [ES2], 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 [KK1], 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 [KK1] are contained in the classes found in [ES2] in a stable range. Furthermore, we prove that the anti-Morita obstructions $[1^{4m+1}]$ for $m\ge 1$ obtained in [ES2] and a hook-type component $[3,1^5]$ detected in [EE] appear in their gap.