临界直径与Vietoris-Rips滤流
Critical Diameters and Vietoris-Rips Filtrations
- Rutgers University（罗格斯大学）
- University of California San Diego（加利福尼亚大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究临界直径与Vietoris-Rips复形拓扑的关系，证明无临界直径时包含映射为同伦等价，刻画了黎曼流形及球面上的临界直径，并给出反例说明弱斜率平稳直径不一定改变同伦类型。
AI中文摘要：
我们将有限构型的临界直径与Vietoris-Rips复形的拓扑联系起来。对于紧致度量空间和$0<r<s$，我们证明当$[r,s)$不包含任何有限标记构型中直径函数具有零弱斜率的直径时，从尺度$r$到尺度$s$的典型包含是同伦等价。在黎曼流形上，弱斜率平稳性蕴含Clarke临界性。在正直径处，当实现直径的距离是光滑的时，两者都等价于一阶平稳性：不存在一个方向使所有这些距离一阶减小。对于闭连通光滑流形，Clarke临界直径谱的Hausdorff维数为零，即使在割迹处也是如此。在实解析情形下，在固定的标记数下它是有限的，并且对所有标记数是可数的。在单位球面$S^m$（$m\geq1$）上，正的非对径弱斜率平稳构型由非零非负平衡应力刻画。最小的正临界直径是$\arccos(-1/(m+1))$，第一个聚点是$\arccos(-1/m)$。我们通过改编自Lovász的球面堆叠构造提升每一个这样的应力，产生平稳直径，随着层数增加，在下一维中从下方逼近原始值。我们还给出了一般度量空间中的例子，表明弱斜率平稳直径不一定对应于Vietoris-Rips复形同伦类型的变化。
英文摘要：
We relate critical diameters of finite configurations to the topology of Vietoris-Rips complexes. For a compact metric space and $0<r<s$, we prove that the canonical inclusion from scale $r$ to scale $s$ is a homotopy equivalence whenever $[r,s)$ contains no diameter of a finite labelled configuration at which the diameter function has zero weak slope. On Riemannian manifolds, weak-slope stationarity implies Clarke criticality. At positive diameter, when the distances realizing the diameter are smooth, both are equivalent to first-order stationarity: the absence of a direction decreasing all these distances to first order. For closed connected smooth manifolds, the Clarke critical diameter spectrum has Hausdorff dimension zero, even at the cut locus. In the real-analytic case, it is finite at each fixed number of labels and countable over all label numbers. On the unit round sphere $S^m$, $m\geq1$, positive nonantipodal weak-slope stationary configurations are characterized by nonzero nonnegative equilibrium stresses. The least positive critical diameter is $\arccos(-1/(m+1))$, and the first accumulation point is $\arccos(-1/m)$. We lift every such stress through a spherical stack construction adapted from Lovász, producing stationary diameters that approach the original value from below in the next dimension as the number of layers increases. We also give example in general metric space showing that weak-slope stationary diameters need not correspond to changes in the homotopy type of Vietoris-Rips complexes.
