The vertex-graded 1-cocycles of the ribbon graph complex RGC1 are one-dimensional and spanned by the graph G1 corresponding to the Enomoto-Satoh trace.
On the Torelli Lie algebra
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We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus $g$: stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $Sp_{2g}(\mathbb{Z})$-representations lying in the centre.
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One-dimensionality of certain cocycles for the detection of the Johnson cokernel
The vertex-graded 1-cocycles of the ribbon graph complex RGC1 are one-dimensional and spanned by the graph G1 corresponding to the Enomoto-Satoh trace.