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.
Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra
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abstract
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.
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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.