One-dimensionality of certain cocycles for the detection of the Johnson cokernel

TL;DR

This paper proves the uniqueness of a specific 1-cocycle in the ribbon graph complex related to the Enomoto-Satoh trace, showing no other such cocycles can detect the Johnson cokernel.

Contribution

It establishes that the ribbon graph complex admits no additional linearly independent 1-cocycles for Johnson cokernel detection beyond the known one.

Findings

No other linearly independent 1-cocycles exist in the ribbon graph complex.

The proof uses vanishing results of top-dimensional cohomology of moduli spaces.

The ES trace is essentially derived from a unique ribbon graph with one vertex and one edge.

Abstract

The Enomoto-Satoh (ES) trace detects the Johnson cokernel, and its 1-cocycle property is important for the proof that the Johnson image is annihilated by the ES trace. Via the natural map from the ribbon graph complex introduced by Merkulov and Willwacher to the Chevalley-Eilenberg complex of the Lie algebra of symplectic derivations, where the Johnson image lives, the ES trace is essentially obtained from the 1-cocycle given by the unique ribbon graph with one vertex and one edge. In this perspective, this ribbon graph is the "universal" version of the ES trace. The main result of this paper is that there are no other (linearly independent) 1-cocycles in the ribbon graph complex, showing that nothing can be found there for the detection of the Johnson cokernel. The proof is done by applying the result of Church-Farb-Putman and Morita-Sakasai-Suzuki, which states that the virtual top-dimensional cohomology of the moduli space of marked Riemann surfaces vanishes.