scholarly journals On Fuzzy Fundamental Groups and Fuzzy Folding of Fuzzy Minkowski Space

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Adeeb G. Talafha ◽  
Sahar M. Alqaraleh ◽  
A. E. El-Ahmady ◽  
Jehad Al Jaraden
Keyword(s):  
Fundamental Group ◽  
Minkowski Space ◽  
Fundamental Groups ◽  
Combinatorial Characterization ◽  
Identity Group ◽  
Fuzzy Identity

In this paper, we studied the relations between new types of fuzzy retractions, fuzzy foldings, and fuzzy deformation retractions, on fuzzy fundamental groups of the fuzzy Minkowski space M ˜ 4 . These geometrical transformations are used to give a combinatorial characterization of the fundamental groups of fuzzy submanifolds on M ˜ 4 . Then, the fuzzy fundamental groups of the fuzzy geodesics and the limit fuzzy foldings of M ˜ 4 are presented and obtained. Finally, we proved a sequence of theorems concerning the isomorphism between the fuzzy fundamental group and the fuzzy identity group.

scholarly journals Unitarization of Loop Group Representations of Fundamental Groups

2007 ◽  
Vol 187 ◽  
pp. 1-33 ◽  
Author(s):  
Josef Dorfmeister ◽  
Hongyou Wu
Keyword(s):  
Fundamental Group ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Loop Group ◽  
Fundamental Groups ◽  
Matrix Functions ◽  
Group Representations ◽  
Constant Mean Curvature Surfaces ◽  
Finite Set

AbstractIn this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

Characterization of small polygroups by their fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 1950047 ◽  
Author(s):  
D. Heidari ◽  
B. Davvaz
Keyword(s):  
Fundamental Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fundamental Groups ◽  
Trivial Group ◽  
Necessary And Sufficient

In this paper, we consider polygroup [Formula: see text] and prove necessary and sufficient conditions such that [Formula: see text] is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say [Formula: see text], classifies in the class [Formula: see text]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [Formula: see text] and [Formula: see text] are 16 and 3, respectively, and the classes [Formula: see text] and [Formula: see text] are singleton.

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

2021 ◽  
Vol 28 (01) ◽  
pp. 119-130
Author(s):  
M.S.M. Asri ◽  
K.B. Wong ◽  
P.C. Wong
Keyword(s):  
Fundamental Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Infinite Order ◽  
Fundamental Groups ◽  
Subgroup Separability

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form [Formula: see text], where [Formula: see text] has infinite order and [Formula: see text] is finite.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

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