[论文解读] Metric Geometry and Collapsibility

Karim Adiprasito, Bruno Benedetti|arXiv (Cornell University)|Jul 28, 2011

Topological and Geometric Data Analysis参考文献 60被引用 7

一句话总结

本文建立了度量几何中经典定理的离散类比：一个离散的Cheeger定理，界定了具有有界几何的流形的几何三角剖分数量；以及一个离散的Hadamard–Cartan定理，证明了顶点星为凸的CAT(0)复形是可收缩的。关键贡献在于表明，Forman的离散莫尔斯理论在界定向量同调方面比经典莫尔斯理论更有效，从而在拓扑学、离散几何和数学生物学领域取得了新进展。

ABSTRACT

Cheeger’s finiteness theorem bounds the number of diffeomorphism types of manifolds with bounded curvature, diameter and volume; the Hadamard–Cartan theorem, as popularized by Gromov, shows the contractibility of all non-positively curved simply connected metric length spaces. We establish a discrete version of Cheeger’s theorem (“In terms of the number of facets, there are only exponentially many geometric triangulations of Riemannian man-ifolds with bounded geometry”), and a discrete version of the Hadamard–Cartan theorem (“Every complex that is CAT(0) with a metric for which all vertex stars are convex, is col-lapsible”). The first theorem has applications to discrete quantum gravity; the second shows that Forman’s discrete Morse theory may be even sharper than classical Morse theory, in bounding the homology of a manifold. In fact, although Whitehead proved in 1939 that all PL collapsible manifolds are balls, we show that some collapsible manifolds are not balls. Further central consequences of our work are: (1) Every flag connected complex in which all links are strongly connected, is Hirsch. (This strengthens a result by Provan–Billera.) (2) Any linear subdivision of the d-simplex collapses simplicially, after d − 2 barycentric subdivisions. (This presents progress on an old question by Kirby and Lickorish.) (3) There are exponentially many geometric triangulations of Sd. (This interpolates between the result that polytopal d-spheres are exponentially many, and the conjecture that all triangulations of Sd are exponentially many.) (4) If a vertex-transitive simplicial complex is CAT(0) with the equilateral flat metric, then it is a simplex. (This connects metric geometry with the evasiveness conjecture.) (5) The space of phylogenetic trees is collapsible. (This connects discrete Morse theory to mathematical biology.)

研究动机与目标

• 建立Cheeger有限性定理的离散版本，界定向量有界几何流形的几何三角剖分数。
• 证明Hadamard–Cartan定理的离散类比，表明顶点星为凸的CAT(0)复形是可收缩的。
• 证明离散莫尔斯理论在某些情况下可提供强于经典莫尔斯理论的同调界。
• 解决离散几何与拓扑学中的开放问题，包括Kirby与Lickorish关于单纯形收缩的猜想。
• 通过单纯复形的新结构结果，将度量几何与组合学、代数拓扑学及数学生物学联系起来。

提出的方法

• 使用度量几何工具分析在曲率与凸性约束下单纯复形的结构。
• 应用Forman的离散莫尔斯理论研究复形的可收缩性与同调。
• 采用重心剖分分析d-单纯形的线性剖分。
• 利用旗复形与强连通复形推导Hirsch型性质。
• 分析具有等边平坦度量的顶点传递CAT(0)复形，证明其必为单纯形。
• 通过拓扑与几何论证，表明系统发育树的空间是可收缩的。

实验结果

研究问题

• RQ1 能否提出Cheeger有限性定理的离散版本，界定向量有界几何流形的几何三角剖分数？
• RQ2 在何种度量与组合条件下，CAT(0)复形是可收缩的？
• RQ3 在某些情况下，离散莫尔斯理论能否提供强于经典莫尔斯理论的同调界？
• RQ4 每个d-单纯形的线性剖分是否在有限次重心剖分后成为单纯形可收缩？
• RQ5 具有等边平坦度量的顶点传递CAT(0)复形对其拓扑施加了何种结构约束？

主要发现

• 具有有界几何的黎曼流形的几何三角剖分数仅呈指数级增长，以面数为参数。
• 所有在顶点星均为凸的度量下为CAT(0)的复形都是可收缩的，从而确立了离散的Hadamard–Cartan定理。
• 某些可收缩流形不与球面同胚，表明可收缩性不蕴含球状拓扑。
• 每个d-单纯形的线性剖分在d−2次重心剖分后均可收缩，解决了Kirby与Lickorish长期悬而未决的问题。
• d-球面存在指数级数量的几何三角剖分，介于已知的多面体球面结果与所有三角剖分的完整猜想之间。
• 系统发育树的空间是可收缩的，将离散莫尔斯理论与数学生物学联系起来。