On the 2-loop part of the Johnson cokernel

TL;DR

This paper investigates the 2-loop component of the Johnson cokernel, extending previous work on the 1-loop part, and introduces a trace map to detect new components in degree 6.

Contribution

It identifies the 2-loop part of the Johnson cokernel and develops a trace map to detect components not seen by the Enomoto-Satoh trace.

Findings

Identified the 2-loop part of the Johnson cokernel in degree 6.

Developed a 2-loop trace map to detect previously unseen components.

Extended the understanding of the Johnson cokernel's structure.

Abstract

We study stable Sp-decompositions of the cokernel of the Johnson homomorphism. Continuing the work of Conant in 2016, which identified the 1-loop part of the Johnson cokernel as the Enomoto-Satoh obstruction, we study the 2-loop part. Using the corresponding 2-loop trace map, we capture all the components of the Johnson cokernels in degree 6 that cannot be detected by the Enomoto-Satoh trace.