Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Susmit Bagchi

doi:10.3390/sym12061039

Abstract

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.

Highlights

The symmetry breaking in any system involves a wide variety of topological excitations and it contains the associated topological constraints
The algebraic as well as topological properties of homotopy decomposition vary depending on the connectedness of the topological space
This paper proposes the homotopy decompositions of path connected fundamental groups into quotient topological spaces

Summary

Introduction

The symmetry breaking in any system involves a wide variety of topological excitations and it contains the associated topological constraints. The preparations of homotopy groups and related decompositions provide meaningful insights into the phases of a system after the symmetry is broken [1]. The homotopy decomposition in the classifying space is constructed considering that the space contains torsion-free groups [2]. The special homotopy classes in a category of based spaces are proposed allowing the decomposition of stable homotopy. The formulation is based on positive filtration of the space and the Toda bracket [3]. The decomposition and related decomposed homotopy types for metrizable LCn spaces are formulated in [4]. Often the homotopy type of original space and the decomposed space are very similar in LCn spaces.

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