Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces for...

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Bibliographic Details
Published in:Symmetry (Basel) 2021-08, Vol.13 (8), p.1421
Main Author: Bagchi, Susmit
Format: Article
Language:English
Subjects:
Algebra
Algebraic Topology
covering spaces
Embedding
Equivalence
fibers
fundamental groups
General Topology
Group theory
Homomorphisms
Mathematics
path-homotopy
topological spaces
Topology
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Summary:The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13081421