String Topology in Dimensions Two and Three

Abstract

Let V denote the vector space with basis the conjugacy classes in the fundamental 4 group of an oriented surface S. In 1986 Goldman [1] constructed a Lie bracket [,] on V. If a and b are conjugacy classes, the bracket [a; b] is defined as the signed sum over intersection points of the conjugacy classes represented by the loop products taken at the intersection points. In 1998 the authors constructed a bracket on higher dimensional manifolds which is part of String Topology [2]. This happened by accident while working on a problem posed by Turaev [3], which was not solved at the time. The problem consisted in characterizing algebraically which conjugacy classes on the surface S are represented by simple closed curves. Turaev was motivated by a theorem of Jaco and Stallings [4,5] that gave a group theoretical statement equivalent to the three dimensional Poincaré conjecture. This statement involved simple conjugacy classes. Recently a number of results have been achieved which illuminate the area around Turaev’s problem. Now that the conjecture of Poincar`e has been solved, the statement about groups of Jaco and Stallings is true and one may hope to find a Group Theory proof. Perhaps the results to be described here could play a role in such a proof. See Sect. 3 for some first steps in this direction.