On the Fundamental Groups of Galois Covering Spaces of the Projective Plane

2004 ◽  
Vol 105 (1) ◽  
pp. 85-105 ◽  
Author(s):  
Makoto Namba ◽  
Hiroyasu Tsuchihashi
Keyword(s):  
Projective Plane ◽  
Fundamental Groups ◽  
Covering Spaces ◽  
Galois Covering

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals On complex line arrangements and their boundary manifolds

2015 ◽  
Vol 159 (2) ◽  
pp. 189-205 ◽  
Author(s):  
V. FLORENS ◽  
B. GUERVILLE-BALLÉ ◽  
M.A. MARCO-BUZUNARIZ
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Fundamental Groups ◽  
Complex Line ◽  
Complex Projective Plane ◽  
Line Arrangement ◽  
Line Arrangements ◽  
Regular Neighbourhood ◽  
Complex Projective

AbstractLet ${\mathcal A}$ be a line arrangement in the complex projective plane $\mathds{C}\mathds{P}^2$. We define and describe the inclusion map of the boundary manifold, the boundary of a closed regular neighbourhood of ${\mathcal A}$, in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

scholarly journals Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2382
Author(s):  
Susmit Bagchi
Keyword(s):  
Multiplicative Group ◽  
Equivalence Classes ◽  
Fundamental Groups ◽  
Local Homeomorphism ◽  
Covering Spaces ◽  
Trivial Group ◽  
Algebraic Forms ◽  
Simply Connected ◽  
Path Homotopy ◽  
Group Structures

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

scholarly journals Calculation of Fundamental Groups via Computational Paths

2021 ◽  
Author(s):  
T. M. L. de Veras ◽  
A. F. Ramos ◽  
R. J. G. B. de Queiroz ◽  
A. G. de Oliveira
Keyword(s):  
Projective Plane ◽  
Term Rewriting ◽  
Fundamental Groups ◽  
Real Projective Plane ◽  
The Real ◽  
Term Rewriting System ◽  
Rewriting System

We address the question as to how to formalise the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. The intention is to demonstrate the use of a term rewriting system in performing computations with these computational paths, establishing equalities between equalities, and further higher equalities, in particular, in the calculation of fundamental groups of surfaces such as the circle, the torus and the real projective plane.

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