arXiv 1905.080492019-05math.GT

关于群$G_{n}^{k}$和$Γ_{n}^{k}$：流形、动力学和不变量的研究

On Groups $G_{n}^{k}$ and $Γ_{n}^{k}$: A Study of Manifolds, Dynamics, and Invariants

Vassily O. Manturov, Denis A. Fedoseev, Seongjeong Kim, Igor M. Nikonov

备注：: 135 pages, 109 figures

摘要：

最近，第一个命名的作者定义了一个2-参数族的群$G_n^k$。这些群可以看作是辫群的类似物。研究$G_n^k$和动力系统之间的联系，发现了以下基本原理：如果描述$n$粒子运动的动力系统具有一个很好的余维1属性，该属性完全由$k$粒子控制，那么这些动力系统允许在$G_{n}^{k}$中具有拓扑不变量值。$G_n^k$群与不同的代数结构有联系。研究的$G_n^k$集团导致，特别是建设的不变量，价值在自由产品的循环群。$G_{n}^{k}$群的所有生成元都是反射，但有许多方法可以增强它们以消除$2$ -挠。后来第一和第四命名的作者介绍和研究了第二族的群体，表示为$Γ_n^k$，这是密切相关的三角形的流形。给定流形的三角剖分空间已被广泛研究。Pachner定理说，任何两个三角形的一个给定的流形可以连接的序列bistellar移动或Pachner移动。当考虑具有固定点数的三角剖分集时，$Γ_n^k$自然出现。引入$Γ_n^k$的方法有两种：几何方法和拓扑方法。第二个可以被认为是流形的“辫子群”，并且是流形的拓扑类型的不变量;以类似的方式，可以构造光滑版本。本文综述了$G_n^k$和$Γ_n^k$理论的基础思想，并对这些群、流形、动力系统、纽结和辫子理论的研究结果作了概述。

英文摘要：

Recently the first named author defined a 2-parametric family of groups $G_n^k$. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups $G_n^k$ and dynamical systems led to the discovery of the following fundamental principle: If dynamical systems describing the motion of $n$ particles possess a nice codimension 1 property governed by exactly $k$ particles, then these dynamical systems admit a topological invariant valued in $G_{n}^{k}$. The $G_n^k$ groups have connections to different algebraic structures. Study of the $G_n^k$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the $G_{n}^{k}$ groups are reflections but there are many ways to enhance them to get rid of $2$-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by $Γ_n^k$, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. Theorem of Pachner says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. $Γ_n^k$ naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing $Γ_n^k$: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a «braid group» of the manifold and is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In the present paper we give a survey of the ideas lying in the foundation of the $G_n^k$ and $Γ_n^k$ theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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