Integration of vector fields on cell complexes and Morse theory
Integration of vector fields on cell complexes and Morse theory
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细胞复合体上矢量场的积分和莫尔斯理论
DOI：
10.1016/j.jmaa.2022.126982
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发表时间：
2023
影响因子：
1.3
通讯作者：
Takeo Nishinou
中科院分区：
文献类型：
--
作者：
Takeo Nishinou
本文研究了多面体复形上的向量场及其相应的轨线。我们研究的向量场是类似的梯度向量场的函数在光滑的情况下。我们的目标是定义一个很好的理论，这样的向量场的轨迹，使他们的集合捕捉多面体复杂的拓扑结构，在经典的莫尔斯理论。由于我们没有假设多面体复形是一个流形，它上的向量场的定义与光滑情形有很大的不同。然而，我们将证明，存在很好的类的功能和度量，使梯度向量场所需的属性。我们的结构依赖于福尔曼的离散莫尔斯理论。特别是，我们使用的函数类是福尔曼离散莫尔斯函数的改进。我们的理论的一个显着特点是，我们的梯度向量场是纯粹从函数和度量定义的光滑的情况下，相反的情况下，离散的莫尔斯理论，我们需要的数据的尺寸的细胞。这使我们能够实现几个有用的结构，这些结构在离散情况下是不可用的。
In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogues of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.
Bestvina–Brady 离散莫尔斯理论和 Vietoris–Rips 复合体
影响因子：
1.7
作者：
M. C. B. Zaremsky
通讯作者：
M. C. B. Zaremsky
关于拓扑和分段线性向量场
DOI：
--
发表时间：
1975
期刊：
影响因子：
--
作者：
R. Stern
通讯作者：
R. Stern