scholarly journals End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds

2005 ◽  
Vol 9 (2) ◽  
pp. 971-990 ◽  
Author(s):  
Robert Myers
Keyword(s):  
Fundamental Groups ◽  
Covering Spaces

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.

Suciu’s ribbon 2-knots with isomorphic group

2020 ◽  
Vol 29 (07) ◽  
pp. 2050053
Author(s):  
Taizo Kanenobu ◽  
Toshio Sumi
Keyword(s):  
Homotopy Groups ◽  
Fundamental Groups ◽  
Covering Spaces ◽  
Isomorphic Group ◽  
Branched Covering ◽  
Trefoil Knot

Suciu constructed infinitely many ribbon 2-knots in [Formula: see text] whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using [Formula: see text]-representations of the fundamental groups of the 2-fold branched covering spaces.

scholarly journals Foldings and monomorphisms

1996 ◽  
Vol 39 (1) ◽  
pp. 51-56
Author(s):  
R. Z. Goldstein
Keyword(s):  
Fundamental Groups ◽  
Free Products ◽  
Covering Spaces ◽  
Folding Process ◽  
Products Of Groups

In this paper we generalize the folding process initiated by Stallings for graphs to a class of generalized covering spaces. These spaces are called pinched coverings or pinched cores, depending on the particular situation. We then apply our generalized folding process to manipulate these spaces into actual coverings. By using elementary homotopy arguments, we can calculate the fundamental groups of these spaces. As a corollary to our main result we obtain a generalization of a result due to Gersten concerning monomorphisms between free products of groups.

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