Moira Chas

Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra

Imagine a collection of closed oriented curves depending on parameters in a smooth d-manifold M. Along a certain locus of configurations strands of the curves may intersect at certain sites in M. At these sites in M the curves may be cut and reconnected in some way. One obtains operators on the set of parametrized collections of closed curves in M. By making the coincidences transversal and compactifying, the operators can be made to act in the algebraic topology of the free loop space of M when M is oriented. The process reveals collapsing sub graph combinatorics like that for removing infinities from Feynman graphs.

The extended Goldman bracket determines intersection numbers for surfaces and orbifolds

In the mid eighties Goldman proved that an embedded closed curve could be isotoped to not intersect a given 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 Goldmans. 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.

Self-Intersection Numbers of Curves on the Punctured Torus

On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded (sharply) by a quadratic function of its combinatorial length (the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses). We show that if a homotopy class has combinatorial length L, then its number of essential self-intersections is bounded by (L − 2)2/4 if L is even, and (L − 1)(L − 3)/4 if L is odd. The classes attaining this bound can be explicitly described in terms of the generators; there are (L − 2)2 + 4 of them if L is even, and 2(L − 1)(L − 3) +8 if L is odd. Similar descriptions and counts are given for classes with self-intersection number equal to one less than the bound. Proofs use both combinatorial calculations and topological operations on representative curves. Computer-generated data are tabulated by counting for each nonnegative integer how many length-L classes have that selfintersection number, for each length L less than or equal to 13. Such experiments led to the results above. Experimental data are also presented for the pair-of-pants surface.

Self-Intersection Numbers of Curves in the Doubly Punctured Plane

We address the problem of computing bounds for the self-intersection number (the minimum number of generic self-intersection points) of members of a free homotopy class of curves in the doubly punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of a set of standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L 2/4+L/2−1, and that when L is even, this bound is sharp; in that case, there are exactly four distinct classes attaining that bound. For odd L we conjecture a smaller upper bound, (L 2−1)/4, and establish it in certain cases in which we show that it is sharp. Furthermore, for the doubly punctured plane, these self-intersection numbers are bounded below, by L/2−1 if L is even, and by (L−1)/2 if L is odd. These bounds are sharp.

String Topology in Dimensions Two and Three

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 Poincare 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.

Self-Intersection Numbers of Length-Equivalent Curves on Surfaces

Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length-equivalent if the lengths of the geodesics in the two classes are equal for every hyperbolic structure on the surface. We show that there are elements in the free group on two generators that are length-equivalent but have different self-intersection numbers. This applies to both the punctured torus and the pair of pants. Our result answers open questions about length-equivalence classes and raises new ones.

Experiments Suggesting That the Distribution of the Hyperbolic Length of Closed Geodesics Sampling by Word Length Is Gaussian

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic in that class. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length on the pair of pants surface. Our experiments strongly suggest that the distribution is normal.