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

TL;DR

This paper compares two different approaches to constructing classes in the Johnson cokernels of surface mapping class groups, showing their relationships and identifying specific classes within their frameworks.

Contribution

It demonstrates that classes from a topological approach are contained within those from a representation theoretic approach and identifies particular classes in their intersection.

Findings

Classes from topological and representation theoretic approaches are related.

Certain obstructions and hook-type classes appear in both frameworks.

The paper clarifies the structure of Johnson cokernels in a stable range.

Abstract

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.