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By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpretthe divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satohobstructions for the surjectivity of the Johnson homomorphisms as some part ofa regular homotopy version of the Turaev cobracket.

A regular homotopy version of the Goldman-Turaev Lie bialgebra, the Enomoto-Satoh traces and the divergence cocycle in the Kashiwara-Vergne problem