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Open AccessArticle

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea
Symmetry 2020, 12(6), 1039; https://doi.org/10.3390/sym12061039
Received: 31 May 2020 / Revised: 18 June 2020 / Accepted: 19 June 2020 / Published: 21 June 2020
The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space. View Full-Text
Keywords: topological spaces; quotient topology; fundamental groups; homotopy; embeddings topological spaces; quotient topology; fundamental groups; homotopy; embeddings
MDPI and ACS Style

Bagchi, S. Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces. Symmetry 2020, 12, 1039.

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