|
| | | | |
| | | |
| Stat |
Members: 3647
Articles: 2'510'585
Articles rated: 2609
05 May 2024 |
|
| | | | |
|
Article overview
| |
|
|
On Groups $G_{n}^{k}$ and $Gamma_{n}^{k}$: A Study of Manifolds, Dynamics, and Invariants | | Vassily O. Manturov
; Denis A. Fedoseev
; Seongjeong Kim
; Igor M. Nikonov
; | | Date: |
20 May 2019 | | Abstract: | Recently the first named author defined a 2-parametric family of groups
$G_n^k$ cite{gnk}. Those groups may be regarded as a certain generalisation 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 one 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,
Coxeter groups and Kirillov-Fomin algebras, to name just a few. Study of the
$G_n^k$ groups led to, in particular, the construction of invariants, valued in
free products of cyclic groups.
Later the first and the fourth named authors introduced and studied the
second family of groups, denoted by $Gamma_n^k$, which are closely related to
triangulations of manifolds.
The spaces of triangulations of a given manifolds have been widely studied.
The celebrated theorem of Pachner~cite{pach} says that any two triangulations
of a given manifold can be connected by a sequence of bistellar moves or
Pachner moves. See also~cite{GelKapZel,nab}; the $Gamma_n^k$ naturally appear
when considering the set of triangulations with the fixed number of points.
There are two ways of introducing this groups: 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, by definition, 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 $Gamma_n^k$ theories and give an overview of recent results in
the study of those groups, manifolds, dynamical systems, knot and braid
theories. | | Source: | arXiv, 1905.8049 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
REGISTER