Ornate necklaces and the homology of the genus one mapping class group

Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection

For an oriented 2-dimensional manifold $Σ$ of genus $g$ with $n$ boundary components the space $\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtration) is described by cyclic words in $H_1(Σ)$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [G. Massuyeau, Formal descriptions of Turaev's loop operations] using Kontsevich integral…