标题： 边界曲面的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，当一个带有非空边界的曲面的基本群中的共轭类由简单闭曲线表示时的条件。 我们证明了以下结论：一个非幂共轭类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有关。 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Jaco和Stallings提出的庞加莱猜想，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的表示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.