标题： 边界曲面上简单闭曲线的代数特征 显示英文标题

标题： An algebraic characterization of simple closed curves on surfaces with boundary

摘要： 我们通过Goldman李代数来描述，当曲面边界非空时，基本群中的哪些共轭类可以通过简单闭曲线表示。 我们证明了以下结论：非幂共轭类X包含嵌入表示当且仅当X与X的三次幂的Goldman李括号为零。 证明使用了组合群论以及Chas对括号的组合描述，此处将其重新表述为Cohen-Lustig算法的说明。 利用Ivanov、Korkmaz和Luo的结果，有推论描述了哪些共轭类的排列与曲面的微分同胚相关。 这个问题的动机来自于二十世纪六十年代的一个群论陈述，该陈述等价于Poincare猜想，由Jaco和Stallings提出，以及二十世纪八十年代Turaev提出的问题。 我们的主要定理实际上根据X与X的三次幂的括号来计算共轭类X的代表的自交点的最小可能数目。 显示英文摘要

摘要： We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permutations of conjugacy classes are related to diffeomorphisms of the surfaces. The problem is motivated by a group theoretical statement from the sixties equivalent to the Poincare conjecture due to Jaco and Stallings and by a question of Turaev from the eighties. Our main theorem actually counts the minimal possible number of self-intersection points of representatives of a conjugacy class X in terms of the bracket of X with the third power of X.