Journal of Topology and Analysis最新文献

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.

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

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.

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.

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.

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$X\subset {ℝ}^{2n}$ contains Sternfeld arrays of arbitrary large size then$X$ is not basic.

The paper provides a simpler proof of this theorem.

Let $F_g$ be a closed orientable surface of genus $g$. A set $\mathrm{\Omega }=\{{\gamma }_1,\dots ,{\gamma }_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a filling system or simply be filling of $F_g$, if $F_g\setminus \mathrm{\Omega }$ is a disjoint union of $b$ topological discs for some $b\ge 1$. A filling system is called minimally intersecting, if the total number of intersection points of the curves is minimum, or equivalently $b=1$. 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 $2g$. Next, we show that for $g\ge 2\text{ and }2\le s\le 2g$ with $(g,s)\ne (2,2)$, there exists a minimally intersecting filling system on $F_g$ of size $s$. Furt

In this paper, we investigate asymptotics for the minimal spanning acycles (MSAs) of the (Alpha)-Delaunay complex on a stationary Poisson process on ${ℝ}^d,d\ge 2$. 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.

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.

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^2(\mathrm{\text{SL}}(2,ℝ))$. 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, $C^{\ast }$-algebras, and Levinson’s theorem.