{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:57:24Z","timestamp":1760237844997,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,21]],"date-time":"2020-06-21T00:00:00Z","timestamp":1592697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym12061039","type":"journal-article","created":{"date-parts":[[2020,6,24]],"date-time":"2020-06-24T07:14:19Z","timestamp":1592982859000},"page":"1039","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces"],"prefix":"10.3390","volume":"12","author":[{"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"134520","DOI":"10.1103\/PhysRevB.95.134520","article-title":"Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases","volume":"95","author":"Higashikawa","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1017\/S0305004100001638","article-title":"A homotopy decomposition for the classifying space of virtually torsion-free groups and applications","volume":"120","author":"Lee","year":"1996","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/1970586","article-title":"The decomposition of stable homotopy","volume":"87","author":"Cohen","year":"1968","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1090\/S0002-9947-1969-0273574-9","article-title":"Homotopy properties of decomposition spaces","volume":"143","author":"Armentrout","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3083","DOI":"10.1090\/S0002-9939-98-04399-8","article-title":"The fundamental group of a compact metric space","volume":"126","author":"Pawlikowski","year":"1998","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s40062-013-0042-7","article-title":"On fundamental groups with the quotient topology","volume":"10","author":"Brazas","year":"2015","journal-title":"J. Homotopy Relat. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.aim.2019.01.043","article-title":"Fundamental groups of locally connected subsets of the plane","volume":"347","author":"Conner","year":"2019","journal-title":"Adv. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"249","DOI":"10.2307\/1969997","article-title":"On the fundamental group of a homogeneous space","volume":"66","author":"Mostow","year":"1957","journal-title":"Ann. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"225","DOI":"10.3390\/sym4010225","article-title":"Classical Knot Theory","volume":"4","author":"Carter","year":"2012","journal-title":"Symmetry"},{"key":"ref_10","first-page":"261","article-title":"On the Connection between the Fundamental Groups of Some Related Spaces","volume":"55","author":"Kampen","year":"1933","journal-title":"Am. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1073\/pnas.45.3.372","article-title":"On the homology and homotopy decomposition of continuous maps","volume":"45","author":"Eckmann","year":"1959","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","first-page":"235","article-title":"Recent Research in Hyperspace Theory","volume":"18","author":"Charatonik","year":"2003","journal-title":"Extr. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/0040-9383(92)90065-P","article-title":"Homotopy Decomposition of Classifying Elementary Abelian Subgroups","volume":"31","author":"Jackowski","year":"1992","journal-title":"Topology"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.top.2003.09.007","article-title":"Homotopy decomposition of a group of symplectomorphisms of S2 \u00d7 S2","volume":"43","author":"Anjos","year":"2004","journal-title":"Topology"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s002290170048","article-title":"A primary construction to topological embeddings and its applications","volume":"Volume 104","author":"Biasi","year":"2001","journal-title":"Manuscripta Mathematica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/S0049-237X(08)71107-8","article-title":"About the axiom of choice","volume":"99","author":"Jech","year":"1977","journal-title":"Stud. Log. Found. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Aguilera, J.P. (2020). Determinate logic and the axiom of choice. Ann. Pure Appl. Log., 171.","DOI":"10.1016\/j.apal.2019.102745"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3305","DOI":"10.1090\/S0002-9947-05-03964-4","article-title":"On Decompositions in Homotopy Theory","volume":"358","author":"Gray","year":"2005","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1016\/j.topol.2009.01.012","article-title":"On the fundamental group of the Sierpinski-gasket","volume":"156","author":"Akiyama","year":"2009","journal-title":"Topol. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1039\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:41:24Z","timestamp":1760175684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,21]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12061039"],"URL":"https:\/\/doi.org\/10.3390\/sym12061039","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,6,21]]}}}