Journal of Topology and Analysis最新文献

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Automorphisms and subdivisions of Helly graphs 赫利图的自动形态和细分

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-05-31 DOI: 10.1142/s179352532450016x

Thomas Haettel

In this paper, we study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graph, in terms of cliques which are intersections of balls.

We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

在本文中，我们研究有限组合维度的 Helly 图，即其注入全域是有限维度的。根据 Lang 的研究成果，我们描述了 Helly 图的注入全域的非常简单的细简细分。我们还给出了一个非常明确的 Helly 图注入全壳的简单模型，即球的交集。我们利用这些细分来证明具有有限组合维度的 Helly 图的任何自动形要么是椭圆形的，要么是双曲形的。此外，每一个这样的双曲自形变在适当的 Helly 细分中都有一个轴，而且它的平移长度是有理的，分母是均匀有界的。

引用次数: 0

Involution generators of the big mapping class group 大映射类群的卷积发电机

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-05-29 DOI: 10.1142/s1793525324500171

Tüli̇n Altunöz, Mehmetci̇k Pamuk, Oğuz Yıldız

Let $S = S (n)$ denote the infinite-type surface with $n$ ends, $n \in ℕ$ , accumulated by genus. For $n \geq 6$ , we show that the mapping class group of $S$ is topologically generated by five involutions. When $n \geq 3$ , it is topologically generated by six involutions.

让 S=S(n) 表示有 n 个端点的无穷型曲面，n∈ℕ，按属累加。当 n≥6 时，我们证明 S 的映射类群由五个渐开线拓扑生成。当 n≥3 时，它由六个渐开线拓扑生成。

引用次数: 0

Packing meets topology 包装与拓扑

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-04-27 DOI: 10.1142/s1793525324500158

Michael H. Freedman

This note initiates an investigation of packing links into a region of Euclidean space to achieve a maximal density subject to geometric constraints. The upper bounds obtained apply only to the class of homotopically essential links and even there seem extravagantly large, leaving much working room for the interested reader.

本论文开始研究如何将链接打包到欧几里得空间的一个区域，以在几何约束条件下达到最大密度。所获得的上界只适用于同位本质链接的类别，即使在该类别中也显得非常大，这给感兴趣的读者留下了很大的工作空间。

引用次数: 0

Structure invariant properties of the hierarchically hyperbolic boundary 分层双曲边界的结构不变特性

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-03-25 DOI: 10.1142/s1793525323500516

Carolyn Abbott, Jason Behrstock, Jacob Russell

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a “maximization” procedure introduced by the first two authors and Durham.

我们证明了层次双曲群边界的几个拓扑和动力学性质与具体的层次双曲结构无关。这是通过证明边界在前两位作者和杜伦提出的 "最大化 "程序下是不变的来实现的。

引用次数: 0

Persistent homotopy groups of metric spaces 度量空间的持久同调群

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-03-22 DOI: 10.1142/s1793525324500018

Facundo Mémoli, Ling Zhou

In this paper, we study notions of persistent homotopy groups of compact metric spaces. We pay particular attention to the case of fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii–Plaut and Barcelo et al. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure which encodes more information than its persistent homology counterpart. We also consider the rationalization of the persistent homotopy groups and by invoking results of Adamaszek–Adams and Serre, we completely characterize them in the case of the circle. Finally, we establish that persistent homotopy groups enjoy stability in the Gromov–Hausdorff sense. We then discuss several implications of this result including that the critical spectrum of Plaut et al. is also stable under this notion of distance.

本文研究了紧凑度量空间的持久同调群概念。我们特别关注基群的情况，通过贝尔斯托夫斯基-普劳特和巴塞洛等人提出的离散基群概念的持久版本，我们获得了对基群的更精确描述。在相当温和的空间假设条件下，我们证明持久基群具有树状结构，它比持久同调群编码了更多信息。我们还考虑了持久同调群的合理化，并通过引用阿达马泽克-亚当斯和塞雷的结果，完全描述了它们在圆情况下的特征。最后，我们确立了持久同调群在格罗莫夫-豪斯多夫意义上的稳定性。然后，我们讨论了这一结果的几种含义，包括普劳特等人的临界谱在这种距离概念下也是稳定的。

引用次数: 0

A simpler proof of Sternfeld’s Theorem 斯特恩菲尔德定理的简单证明

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-03-11 DOI: 10.1142/s1793525324500080

S. Dzhenzher

In Sternfeld’s work on Kolmogorov’s Superposition Theorem appeared the combinatorial–geometric notion of a basic set and a certain kind of arrays. A subset $X \subset ℝ^{n}$ is basic if any continuous function $X \to ℝ$ could be represented as the sum of compositions of continuous functions $ℝ \to ℝ$ and projections to the coordinate axes.

The definition of a Sternfeld array is presented in this paper.

Sternfeld’s Arrays Theorem.If a closed bounded subset $X \subset ℝ^{2 n}$ contains Sternfeld arrays of arbitrary large size then $X$ is not basic.

The paper provides a simpler proof of this theorem.

在斯特恩费尔德关于科尔莫戈罗夫叠加定理的研究中，出现了基本集和某种数组的组合几何概念。如果任何连续函数 X→ℝ 都可以表示为连续函数 ℝ→ℝ 的组合和对坐标轴的投影，那么子集 X⊂ℝn 就是基本集。斯特恩费尔德数组定理.如果一个封闭的有界子集X⊂ℝ2n包含任意大的斯特恩费尔德数组，那么X不是基本的.本文提供了这个定理的一个更简单的证明.

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引用次数: 0

Filling systems on surfaces 表面填充系统

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-03-06 DOI: 10.1142/s1793525324500055

Shiv Parsad, Bidyut Sanki

Let <math altimg="eq-00001.gif" display="inline" overflow="scroll"><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub></math> be a closed orientable surface of genus <math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>g</mi></math>. A set <math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi mathvariant="normal">Ω</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo stretchy="false">}</mo></math> of pairwise non-homotopic simple closed curves on <math altimg="eq-00004.gif" display="inline" overflow="scroll"><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub></math> is called a filling system or simply be filling of <math altimg="eq-00005.gif" display="inline" overflow="scroll"><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub></math>, if <math altimg="eq-00006.gif" display="inline" overflow="scroll"><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub><mo stretchy="false">∖</mo><mi mathvariant="normal">Ω</mi></math> is a disjoint union of <math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>b</mi></math> topological discs for some <math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>b</mi><mo>≥</mo><mn>1</mn></math>. A filling system is called minimally intersecting, if the total number of intersection points of the curves is minimum, or equivalently <math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>b</mi><mo>=</mo><mn>1</mn></math>. The size of a filling system is defined as the number of its elements. We prove that the maximum possible size of a minimally intersecting filling system is <math altimg="eq-00010.gif" display="inline" overflow="scroll"><mn>2</mn><mi>g</mi></math>. Next, we show that for <math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>g</mi><mo>≥</mo><mn>2</mn><mstyle><mtext> and </mtext></mstyle><mn>2</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mn>2</mn><mi>g</mi></math> with <math altimg="eq-00012.gif" display="inline" overflow="scroll"><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≠</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>, there exists a minimally intersecting filling system on <math altimg="eq-00013.gif" display="inline" overflow="scroll"><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub></math> of size <math altimg="eq-00014.gif" display="inline" overflow="scroll"><mi>s</mi></math>. Furt

如果 Fg∖Ω 是某个 b≥1 的 b 个拓扑圆盘的不相交联盟，则 Fg 上的一对非同调简单闭合曲线集合 Ω={γ1,...,γs} 称为填充系统或简称为 Fg 的填充。如果曲线的交点总数最小，或等价于 b=1，则填充系统称为最小相交。填充系统的大小定义为其元素的个数。我们证明最小相交填充系统的最大可能大小为 2g。接着，我们证明了对于 g≥2 且 2≤s≤2g 且 (g,s)≠(2,2) 的情况，在 Fg 上存在一个大小为 s 的最小相交填充系统。对于 g≥2，我们证明了对于大小为 s 的最小相交填充系统 Ω，几何相交数满足 max{i(γi,γj)|i≠j}≤2g-s+1，并且对于每个这样的 s，存在一个最小相交填充系统 Ω={γ1,...,γs}，使得 max{i(γi,γj)|i≠j}=2g-s+1。

引用次数: 0

Central limit theorem for euclidean minimal spanning acycles 欧几里得最小跨度循环的中心极限定理

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-02-29 DOI: 10.1142/s1793525323500590

Primoz Skraba, D. Yogeshwaran

In this paper, we investigate asymptotics for the minimal spanning acycles (MSAs) of the (Alpha)-Delaunay complex on a stationary Poisson process on $ℝ^{d}, d \geq 2$ . MSAs are topological (or higher-dimensional) generalizations of minimal spanning trees. We establish a central limit theorem (CLT) for total weight of the MSA on a Poisson Alpha-Delaunay complex. Our approach also allows us to establish CLTs for the sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of MSAs which proceeds by establishing suitable chain maps and uses matroidal properties of MSAs. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.

在本文中，我们研究了 ℝd,d≥2 上静止泊松过程的 (Alpha)-Delaunay 复数的最小跨循环（MSA）的渐近性。MSA 是最小生成树的拓扑（或高维）概括。我们建立了泊松 Alpha-Delaunay 复数上 MSA 总重的中心极限定理（CLT）。我们的方法还允许我们建立德劳内复合体持久图中出生时间和寿命之和的中心极限定理。我们证明的关键在于展示所谓的 MSA 弱稳定，即通过建立合适的链映射并利用 MSA 的矩阵性质来实现。与通过渗流理论估计对欧氏最小生成树的弱稳定证明不同，我们的弱稳定证明是代数性质的，即使在最小生成树的情况下也提供了另一种证明。

引用次数: 0

A solvable extended logarithm of the Johnson homomorphism 约翰逊同构的可解扩展对数

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-02-22 DOI: 10.1142/s1793525324500079

Takefumi Nosaka

Concerning Johnson’s homomorphism from the Torelli group, there are previous works to define a logarithm of the homomorphism, and give some extension of the logarithm. This paper considers exponential solvable elements in the mapping class group of a surface, and defines the logarithms of such elements.

关于托雷利群的约翰逊同态，以前有一些著作定义了同态的对数，并给出了对数的一些扩展。本文考虑了曲面映射类群中的指数可解元素，并定义了这些元素的对数。

引用次数: 0

Scattering theory and an index theorem on the radial part of SL(2, ℝ) 散射理论和 SL(2, ℝ) 径向部分的指数定理

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Topology and Analysis

Pub Date : 2024-02-17 DOI: 10.1142/s179352532450002x

H. Inoue, S. Richard

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^{2} (SL (2, ℝ))$ . After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, $C^{*}$ -algebras, and Levinson’s theorem.

我们介绍了作用于 L2(SL(2,ℝ) 中径向函数的卡西米尔算子的谱和散射理论。）经过适当分解后，这些研究包括研究作用于半线的微分算子族。对于这些算子，可以根据高斯超几何函数找到解析量、谱密度和莫勒波算子的明确表达式。此外，还引入并讨论了一个指数定理。由此产生的等式一般称为列文森定理，它将超几何函数的各种渐近行为联系起来。这项研究首次尝试将群论、特殊函数、散射理论、C∗-代数和列文森定理联系起来。

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引用次数: 0

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