Addendum to "The Johnson cokernel and the Enomoto-Satoh invariant": The ES-trace detects all top-level partitions

TL;DR

This paper precisely calculates the top-level component of the Johnson cokernel, showing it is isomorphic to a space of coinvariants and confirming a key conjecture related to the Enomoto-Satoh invariant.

Contribution

It provides an exact computation of the top-level Johnson cokernel component, linking it to coinvariants and confirming a conjecture from prior work.

Findings

The top-level Johnson cokernel component is isomorphic to a space of coinvariants.

The isomorphism is induced by the Enomoto-Satoh trace map.

The result confirms Conjecture 7.2 from the previous paper.

Abstract

The degree $d$ part of the cokernel $\mathsf C_d$ of the Johnson homomorphism decomposes into irreducible $\mathrm{SP}$-modules indexed by partitions of $d-2r$ for $r\geq 0$: $$\mathsf C_d\cong \mathsf C_d(d)\oplus \mathsf C_d(d-2)\oplus\cdots.$$ In this addendum we calculate $\mathsf{C}_d(d)$ precisely: it is isomorphic to the $\mathrm{GL}(V)$-decomposition of a space of coinvariants $(V^{\otimes d})_{D_{2d}}$, and the isomorphism is induced by Enomoto and Satoh's trace map. This establishes Conjecture 7.2 of the paper "The Johnson Cokernel and the Enomoto-Satoh invariant."