A Note on the Construction of Simply-Connected 3-Manifolds as Branched Covering Spaces of S 3

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

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Interactive Design and Visualization of Branched Covering Spaces

IEEE Transactions on Visualization and Computer Graphics ◽

10.1109/tvcg.2017.2744038 ◽

2018 ◽

Vol 24 (1) ◽

pp. 843-852 ◽

Cited By ~ 1

Author(s):

Lawrence Roy ◽

Prashant Kumar ◽

Sanaz Golbabaei ◽

Yue Zhang ◽

Eugene Zhang

Keyword(s):

Interactive Design ◽

Covering Spaces ◽

Branched Covering

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Constructions of two-fold branched covering spaces

Pacific Journal of Mathematics ◽

10.2140/pjm.1986.125.415 ◽

1986 ◽

Vol 125 (2) ◽

pp. 415-446 ◽

Cited By ~ 11

Author(s):

José Montesinos-Amilibia ◽

Wilbur Whitten

Keyword(s):

Covering Spaces ◽

Branched Covering

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Cobordism of branched covering spaces

Pacific Journal of Mathematics ◽

10.2140/pjm.1980.87.335 ◽

1980 ◽

Vol 87 (2) ◽

pp. 335-345 ◽

Cited By ~ 3

Author(s):

Hugh Hilden ◽

Robert D. Little

Keyword(s):

Covering Spaces ◽

Branched Covering

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Homology Invariants

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-057-6 ◽

1978 ◽

Vol 30 (03) ◽

pp. 655-670 ◽

Cited By ~ 5

Author(s):

Richard Hartley ◽

Kunio Murasugi

Keyword(s):

Homology Group ◽

Covering Space ◽

Simple Relationship ◽

Cyclic Covering ◽

Covering Spaces ◽

Homology Groups ◽

Branched Covering ◽

The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

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

Symmetry ◽

10.3390/sym13122382 ◽

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.

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