{"title":"Simple homotopy of flag simplicial complexes and contractible contractions of graphs","authors":"Anton Dochtermann , Takahiro Matsushita","doi":"10.1016/j.topol.2025.109326","DOIUrl":"10.1016/j.topol.2025.109326","url":null,"abstract":"<div><div>In his work on molecular spaces, Ivashchenko introduced the notion of an <span><math><mi>I</mi></math></span>-contractible transformation on a graph <em>G</em>, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the <span><math><mi>I</mi></math></span>-homotopy type of a graph, and showed that <span><math><mi>I</mi></math></span>-contractible transformations preserve the simple homotopy type of <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, the clique complex of <em>G</em>. In other work, Boulet, Fieux, and Jouve introduced the notion of <em>s</em>-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that <em>s</em>-homotopy preserves <span><math><mi>I</mi></math></span>-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs <em>G</em> and <em>H</em> are <span><math><mi>I</mi></math></span>-homotopy equivalent if and only if <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> are simple homotopy equivalent. We also show that a finite graph <em>G</em> is <span><math><mi>I</mi></math></span>-contractible if and only if <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109326"},"PeriodicalIF":0.6,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The answers to two problems on maximal point spaces of domains","authors":"Xiaoyong Xi , Chong Shen , Dongsheng Zhao","doi":"10.1016/j.topol.2025.109340","DOIUrl":"10.1016/j.topol.2025.109340","url":null,"abstract":"<div><div>A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space <span><math><mtext>Max</mtext><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of a domain <em>P</em> (with the relative Scott topology). We first construct an example to show that the set of maximal points of an ideal domain <em>P</em> need not be a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-set in the Scott space Σ<em>P</em>, thereby answering an open problem from Martin (2003). In addition, Bennett and Lutzer (2009) asked whether <em>X</em> and <em>Y</em> are domain-representable if their product space <span><math><mi>X</mi><mo>×</mo><mi>Y</mi></math></span> is domain-representable. This problem was first solved by Önal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109340"},"PeriodicalIF":0.6,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143641920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some properties of selectively star-ccc spaces","authors":"Yuan Sun","doi":"10.1016/j.topol.2025.109339","DOIUrl":"10.1016/j.topol.2025.109339","url":null,"abstract":"<div><div>In 2013 <span><span>[2]</span></span>, Aurichi introduced a topological property named selectively ccc that can be viewed as a selective version of the countable chain condition (CCC). Later, Bal and Kočinac in <span><span>[3]</span></span> extended Aurichi's work and defined the star version of the selectively ccc property called selectively <em>k</em>-star-ccc. The aim of this paper is twofold. Firstly, we establish connections between the selectively <em>k</em>-star-ccc properties, the chain conditions and other star-Lindelöf properties. Secondly, some examples are presented to solve questions raised by Xuan and Song in <span><span>[12]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109339"},"PeriodicalIF":0.6,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some results on topological MV-algebras","authors":"Leyao Yin , Lihong Xie , Jiang Yang","doi":"10.1016/j.topol.2025.109328","DOIUrl":"10.1016/j.topol.2025.109328","url":null,"abstract":"<div><div>First of all, in this paper, we give an equivalent characterization for a finite topological MV-algebra satisfying the open condition. In addition, we investigate the topological isomorphism theorems and other related results of topological MV-algebras with the open condition. Then we study the initial topology of topological MV-algebras. Moreover, we introduce the concept of proto-MV-algebras and two operations ∔ and ′ on the family <span><math><mi>ω</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of set-theoretic filters on proto-MV-algebra <em>A</em>. These operations make <span><math><mo>(</mo><mi>ω</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mo>∔</mo><mo>,</mo><mo>′</mo><mo>,</mo><mo>↑</mo><mn>0</mn><mo>)</mo></math></span> a proto-MV-algebra. Especially, we establish a relationship between the category <span><math><mi>PMV</mi></math></span> and the category <span><math><mi>TPMV</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109328"},"PeriodicalIF":0.6,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Lindelöf property of lexicographic products","authors":"Yasushi Hirata , Nobuyuki Kemoto","doi":"10.1016/j.topol.2025.109338","DOIUrl":"10.1016/j.topol.2025.109338","url":null,"abstract":"<div><div>The Lindelöf property of lexicographic products of GO-spaces is characterized.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109338"},"PeriodicalIF":0.6,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A fuzzy topological model of hemimetric-based fuzzy rough set and some applications to digital image processing","authors":"Xinyue Han , Wei Yao , Chang-Jie Zhou","doi":"10.1016/j.topol.2025.109327","DOIUrl":"10.1016/j.topol.2025.109327","url":null,"abstract":"<div><div>This paper aims to present a fuzzy topological model of hemimetric-based fuzzy rough set by introducing the neighborhood-controlled fuzzy rough approximation operators as the fuzzy topological operators. Although the related fuzzy upper/lower rough approximation operators are no longer idempotent, they form a Galois adjoint pair, which makes their compositions idempotent. The composition of upper-lower operators is called the closing operator, which is a closure operator on the fuzzy power set; and that of lower-upper one is called the opening operator, which is an interior operator on the fuzzy power set. Results show that these two operators can be applied to hole filling, fingerprint cleaning and noise reduction in digital image processing.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109327"},"PeriodicalIF":0.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Double homology and wedge-decomposable simplicial complexes","authors":"Carlos Gabriel Valenzuela Ruiz, Donald Stanley","doi":"10.1016/j.topol.2025.109319","DOIUrl":"10.1016/j.topol.2025.109319","url":null,"abstract":"<div><div>We show a wedge-decomposable simplicial complex has associated double homology <span><math><mi>Z</mi><mo>⊕</mo><mi>Z</mi></math></span> in bidegrees <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109319"},"PeriodicalIF":0.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Uroš A. Colović, Milica Jovanović , Branislav I. Prvulović
{"title":"Cohomology rings of oriented Grassmann manifolds G˜2t,4","authors":"Uroš A. Colović, Milica Jovanović , Branislav I. Prvulović","doi":"10.1016/j.topol.2025.109318","DOIUrl":"10.1016/j.topol.2025.109318","url":null,"abstract":"<div><div>We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msup><mo>,</mo><mn>4</mn></mrow></msub></math></span> as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gröbner basis for that ideal, which we then use to exhibit an additive basis for <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msup><mo>,</mo><mn>4</mn></mrow></msub><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109318"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Properties of equi-Baire 1 and equi-Lebesgue families of functions","authors":"Marek Balcerzak , Ľubica Holá , Dušan Holý","doi":"10.1016/j.topol.2025.109317","DOIUrl":"10.1016/j.topol.2025.109317","url":null,"abstract":"<div><div>We study several properties of equi-Baire 1 families of functions between metric spaces. We consider the related equi-Lebesgue property for such families. We examine the behavior of equi-Baire 1 and equi-Lebesgue families with respect to pointwise and uniform convergence. In particular, we obtain a criterion for a choice of a uniformly convergent subsequence from a sequence of functions that form an equi-Baire 1 family, which solves a problem posed in <span><span>[3]</span></span>. Finally, we discuss the notion of equi-cliquishness and relations between equi-Baire 1 families and sets of equi-continuity points.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109317"},"PeriodicalIF":0.6,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Handle numbers of guts of sutured manifolds and nearly fibered knots","authors":"Kenneth L. Baker , Fabiola Manjarrez-Gutiérrez","doi":"10.1016/j.topol.2025.109316","DOIUrl":"10.1016/j.topol.2025.109316","url":null,"abstract":"<div><div>Extending Haken's Theorem to product annuli and disks for Heegaard splittings of sutured manifolds, we show that the handle number of an irreducible sutured manifold equals the handle number of its guts. We further show that reduced sutured manifolds with torus boundary contained in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> fall in to three types that generalize the three models of guts of knots that are nearly fibered in the instanton or Heegaard Floer sense. In conjunction with these results and another concerning uniqueness of incompressible Seifert surfaces, we show that while many nearly fibered knots have handle number 2 and a unique incompressible Seifert surface, some have handle number 4 and others have extra incompressible Seifert surfaces. Examples of nearly fibered knots with non-isotopic incompressible Seifert surfaces are exhibited.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109316"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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