Geometric intersections of loops on surfaces

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.

A geometric property of closed geodesics on hyperbolic surfaces

We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics

Homeomorphisms and intersection number

We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorphisms of PML(S) that preserve the zero sets of these intersection functions.

圆柱端模空间的亏格和几何交的界

Bounds on genus and geometric intersections from cylindrical end moduli spaces

本文给出了一种计算具有第二正贝蒂数$b_2^+(X)=1$的光滑四流形$X$中正自相交的同调类的光滑亏格的下界的方法.本文研究了Seiberg-Witten方程在圆柱端流形上的解，该流形是这类曲面的补曲面。其结果可以表示为一个广义的附加不等式的形式。得到的界仅依赖于流形的有理同调类型，并包括作为特例的Thom猜想。我们推广了这种方法，得到了流形中正自相交的$n$代数不相交曲面与$b_2^+(X)=n$的交点个数的下界。

Computing the Geometric Intersection Number of Curves

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algorithms to (1) compute the geometric intersection number of $c$ in $O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+\ell^4)$ time, (3) decide if the geometric intersection number of $c$ is zero, i.e. if $c$ is hom…

Loops in surfaces and star-fillings

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the surfaces by certain graphs.

自由群曲线复形的几何相交数及其类似

Geometric Intersection Number and analogues of the Curve Complex for free groups

对于有限秩的自由群$N \geq 2$，我们构造了一个规范的Bonahon型连续和$Out(F_N)$不变的\emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \]这里$\bar{cv}(F_N)$是非投影化的Culler-Vogtmann外层空间$cv(F_N)$在等变Gromov-Hausdorff收敛拓扑中（或者等价地，在长度函数拓扑中）的闭包。已知$\bar{cv}(F_N)$由$F_N$在$\mathbb R$ -树上的所有\emph{very small}极小等距作用组成. $\bar{cv}(F_N)$的投影化提供了一个自由群类似于瑟斯顿的泰希米勒空间的紧化。作为应用，利用由交形式确定的\emph{intersection graph}，我们证明了自由群背景下曲线复形的几个自然类似物具有无穷大直径。

Effective mapping class group dynamics II: Geometric intersection numbers

We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the bou…

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 Turaev's Virtual String Cobracket

In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings. We show that $μ$ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev's virtual string cobracket. We use Turaev's based matrices to describe strings $α$ such that $μ$ gives a formula fo…

Distances and intersections of curves

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the curve graphs and the mapping class groups.

球面在球面中的交及Arf不变量的新几何意义

The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant

设S^3_i是嵌入在5-球面S^5（i= 1，2）中的3-球面。设S^3_1和S^3_2横切。那么S^3_1和S^3_2的交点C是一个不相交的圆的集合。因此我们得到一对1-环，C在S^3_i（i= 1，2）中，以及一对3-纽结，S^3_i在S^5（i= 1，2）中。反之，设（L_1，L_2）是一对1-链环，（X_1，X_2）是一对3-纽结.很自然地要问一个问题，一对1-链环（L_1，L_2）是否可以作为上述3-结点X_1和X_2的交集而得到。我们对这个问题给出了完整的回答。我们的答案给出了一个新的几何意义的Arf不变量的1-链接。设f是S^3到S^5的光滑横浸入。则自相交C由二重点组成。假设C是S^5中的一个圆。则S^3中的f^{-1}（C）是1-纽结或2-分量1-链环。还有一个类似的实现问题。我们对这个问题给出了完整的回答。

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。

Intersections of curves on surfaces and their applications to mapping class groups

We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson homomorphism.

Infinitesimal Liouville currents, cross-ratios and intersection numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichmüll…

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…