Simple Loops on Surfaces and Their Intersection Numbers

Formal descriptions of Turaev's loop operations

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[π]$ of the fundamental group $π$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra $\mathbb{Q}[π]$ is completed with respect to powers of its augmentation ideal and is app…

A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface

Goldman and Turaev constructed a Lie bialgebra structure on the free $\mathbb{Z}$-module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket $Δ(α)$ is zero if and only if $α$ is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunately, false. We define an operation $μ$ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through $μ$, so we can view $μ$ as a generalization of $Δ$. We show that Turaev's conjecture holds when $Δ$ is replaced with $μ$. We also…

管状群的可公度性与拟等距分类：单顶点单环情形

Commensurability and quasi-isometry classification for one vertex one loop tubular groups

本文对单顶点单环管状群按边映射交数分类：非零交数时全体可公度且拟等距；零交数时有两个拟等距类和无穷可公度类，两类间不拟等距。

Bounds on the number of non-simple closed geodesics on a surface

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given weak conditions on self-intersection number.

具有指定成对相交数的曲面上的曲线

Curves on surfaces with prescribed pairwise intersection numbers

本文研究在环面及最小亏格曲面上构造具有指定成对相交数的简单闭曲线系统的问题，给出了环面情形下可实现性的充分必要条件，并对一般最小亏格实现问题提供了部分结果。

Product and coproduct in string topology

Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We prove the following new geometric property of the dual homology coproduct: the nonvanishing of the k-th iterate of the coproduct on a homology class ensures the existence of a loop with a (k+1)-fold self-intersection in every representative of the class. For spheres and projective…

Splittings of groups and intersection numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.

通过相交看待曲线的拓扑视角

A topological viewpoint on curves via intersection

本文研究闭曲线能否由相交次数确定，构造并分析 $k$-等价曲线，揭示其与简单曲线相交的性质，并以定量方法联系 Otal 定理。

Geometric intersection in representations of mapping class groups of surfaces

We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem of detecting the geometric intersection between separating simple closed curves in a representation.

Minimal intersection of curves on surfaces

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a s…

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without exp…