含嵌入弧的三流形理想三角剖分的一种演算

Ideal Triangulation and Disk Unfolding of a Singular Flat Surface

An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length $\leq L$ connecting them, where $L$ is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics wh…

Ideally, all infinite type surfaces can be triangulated

We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.

Surgery calculus for classical $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons theory

Classical $\operatorname{SL}_2(\mathbb{C})$-Chern-Simons theory assigns a $3$-manifold $M$ with representation $ρ: π_1(M) \to \operatorname{SL}_2(\mathbb{C})$ its complex volume $\operatorname{V}(M, ρ) \in \mathbb{C} / 2 π^2 i \mathbb{Z}$, with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing $\operatorname{V}$ using a triangulation. In this paper we show how to compute $\operatorname{V}(M, L, ρ)$ directly from a surgery diagram for $M$ a compact oriented $3$-manifold with torus boundary components, embedded cusps $L$, and represe…

4-流形中的弧族与配置空间的映射

Families of Arcs in 4-Manifolds and Maps of Configuration Spaces

本文在4-流形中构造了固定边界的3参数嵌入弧族，利用嵌入微积分和图示框架分析其在Taylor逼近中的平凡性，并用Bousfield-Kan谱序列计算有理同伦群，扩展了Budney和Gabai的相关工作。

组合纤维丛与纤维化PL-同胚的分裂

Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

对于紧PL流形X，我们关联一个范畴T（X）。T（X）的对象都是X型的组合流形，态射是组合集合。证明了同伦等价BT（X）\approx BPL（X）成立，其中PL（X）是PL-同胚的单群.因此空间BT（X）是BPL（X）的一个典范可数（作为CW-复形）模型。结果得到了以PL多面体B为基底，以纤维X为基底的PL纤维丛的函子纯组合模型。这样的模型看起来像B的某个三角剖分K的T（X）-着色。K的顶点被T（X）的对象着色，弧被态射着色，使得由K的2-骨架产生的图是交换的。与经典的几何拓扑结果相比，我们得到了小维实格拉斯曼的组合模型：BT（S^{n-1}）\approx BO（n），其中n= 1，2，3，4.在B \PL（X）的相似模型上的一系列结果中证明了这一结果。特别注意主要的非紧情形X=R^n以及组合流形的切丛和高斯函子。使证明成为可能的技巧是关于“纤维同胚的分裂”的引理的集合，这是关于合痕的分裂的民间传说引理的推广。