In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldman's. The arguments are purely topological, or based on elementary ideas from hyperbolic geometry.
These results are intended to be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three manifolds. The recognition is based on the structure of the String Topology bracket of three manifolds.
Partitions of cyclic words and Goldman-Turaev Lie bialgebra
由定向曲面上的闭曲线的非平凡同伦类集合生成的自由$\mathbb{Z}$ -模通过两种运算（Goldman括号和Turaev余括号）具有李双代数的结构. M. Chas给出了这两个操作的组合重新定义，通过自然识别的同伦类的闭曲线的表面上的循环字的生成器和他们的逆的基本群的表面。我们提出了一种新的方法，给出了一个组合定义的括号和cobracket，侧重于循环词的分区所提供的信息。
Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection
对于具有$n$边界分支的亏格为$g$的定向二维流形$Σ$，空间$\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$具有Goldman-Turaev李双代数结构，该双代数结构由曲线的交和自交定义.它的相关分次（在自然过滤下）由$H_1(Σ)$中的循环词描述，并带有项链Schedler李双代数的结构。在亏格为零的情况下，这两种结构之间的同构已在[G. Massuyeau，使用Kontsevich积分的Turaev循环操作的形式描述[A. Alekseev，N. Kawazumi，Y. Kuno和F. Naef，Goldman-Turaev Lie bibalgebra in genus zero and the Kashiwara-Vergne problem]使用Kashiwara-Vergne问题的解。本文给出了$\mathbb{C}$上这一同构的初等证明。它使用$\mathbb{C}\backslash\{ z_1, \dots z_n\}$上的Knizhnik-Zamolodchikov连接。李括号同构的证明是Hitchin经典结果的一个版本。令人惊讶的是，类似的证明也适用于cobrackets。
