Topology and its Applications最新文献

{"title":"Linear and smooth oriented equivalence of orthogonal representations of finite groups","authors":"Luis Eduardo García-Hernández ,&nbsp;Ben Williams","doi":"10.1016/j.topol.2025.109292","DOIUrl":"10.1016/j.topol.2025.109292","url":null,"abstract":"<div><div>Let Γ be a finite group. We prove that if <span><math><mi>ρ</mi><mo>,</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>Γ</mi><mo>→</mo><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> are two representations that are conjugate by an orientation-preserving diffeomorphism of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, then they are conjugate by an element of <span><math><mi>SO</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>. In the process, we prove that if <span><math><mi>G</mi><mo>⊂</mo><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> is a finite group, then exactly one of the following is true: the elements of <em>G</em> have a common invariant 1-dimensional subspace in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>; some element of <em>G</em> has no invariant 1-dimensional subspace; or <em>G</em> is conjugate to a specific group <span><math><mi>K</mi><mo>⊂</mo><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> of order 16.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"367 ","pages":"Article 109292"},"PeriodicalIF":0.6,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561853","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}

{"title":"Topological and dimensional properties of univoque bases in double-base expansions","authors":"Yuecai Hu ,&nbsp;Rafael Alcaraz Barrera ,&nbsp;Yuru Zou","doi":"10.1016/j.topol.2025.109294","DOIUrl":"10.1016/j.topol.2025.109294","url":null,"abstract":"<div><div>Given two real numbers <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> with <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><mn>1</mn></math></span> satisfying <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, we call a sequence <span><math><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span> a <span><math><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span><em>-expansion</em> or a <em>double-base expansion</em> of a real number <em>x</em> if<span><span><span><math><mi>x</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>⋯</mo><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>.</mo></math></span></span></span> When <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>q</mi></math></span>, the set of <em>univoque bases</em> is given by the set of <em>q</em>'s such that <span><math><mi>x</mi><mo>=</mo><mn>1</mn></math></span> has exactly one <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-expansion. The topological, dimensional and symbolic properties of such sets and their corresponding sequences have been intensively investigated. In our research, we study the topological and dimensional properties of the set of univoque bases for double-base expansions. This problem is more complicated, requiring new research strategies. Several new properties are uncovered. In particular, we show that the set of univoque bases in the double base setting is a meagre set with full Hausdorff dimension.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"366 ","pages":"Article 109294"},"PeriodicalIF":0.6,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551335","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":"Singular homology of roots of unity","authors":"Nikola Milićević","doi":"10.1016/j.topol.2025.109291","DOIUrl":"10.1016/j.topol.2025.109291","url":null,"abstract":"<div><div>We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"366 ","pages":"Article 109291"},"PeriodicalIF":0.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509248","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":"Many subalgebras of P(ω)/fin","authors":"Klaas Pieter Hart","doi":"10.1016/j.topol.2025.109280","DOIUrl":"10.1016/j.topol.2025.109280","url":null,"abstract":"<div><div>In answer to a question on Mathoverflow we show that the Boolean algebra <span><math><mi>P</mi><mo>(</mo><mi>ω</mi><mo>)</mo><mo>/</mo><mrow><mi>fin</mi></mrow></math></span> contains a family <span><math><mo>{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>:</mo><mi>X</mi><mo>⊆</mo><mi>c</mi><mo>}</mo></math></span> of subalgebras with the property that <span><math><mi>X</mi><mo>⊆</mo><mi>Y</mi></math></span> implies <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span> is a subalgebra of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and if <span><math><mi>X</mi><mo>⊈</mo><mi>Y</mi></math></span> then <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span> is not embeddable into <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>. The proof proceeds by Stone duality and the construction of a suitable family of separable zero-dimensional compact spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"366 ","pages":"Article 109280"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509249","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}

{"title":"Completely Scott closed set and its applications","authors":"Licong Sun,&nbsp;Bin Pang","doi":"10.1016/j.topol.2025.109283","DOIUrl":"10.1016/j.topol.2025.109283","url":null,"abstract":"<div><div>In this paper, we propose a concept of completely Scott closed sets and use it to study links between convex spaces and continuous lattices. Firstly, we take three equivalent approaches to construct a convex space from a continuous lattice. Secondly, we construct an adjunction between the category of convex spaces and the opposite category of continuous lattices via completely Scott closed sets. This adjunction exactly induces the concept of sober convex spaces which gives rise to a categorical duality between them and algebraic lattices. Finally, we prove that completely Scott closed sets form a monad over the category of convex spaces and obtain an isomorphism between the category of sober convex spaces and the Eilenberg–Moore category of this monad.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109283"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480602","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":"Representations of branched twist spins with a non-trivial center of order 2","authors":"Mizuki Fukuda","doi":"10.1016/j.topol.2025.109284","DOIUrl":"10.1016/j.topol.2025.109284","url":null,"abstract":"<div><div>A branched twist spin is a 2-knot consisting of exceptional orbits and fixed points of a circle action on the four sphere. It is a generalization of the twist spun knot, and its knot group is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain relator. In this paper, we study on <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-representation for a branched twist spin. For the latter case, we determine the number of 4<em>k</em>-ordered dihedral group representations of branched twist spins. As an application, we can show non-equivalence between two branched twist spins by counting dihedral representations of their knot groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109284"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480603","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":"Neighborhood system and categorical properties of quasitopological vector spaces","authors":"Zhongqiang Yang ,&nbsp;Yajing Fang ,&nbsp;Qiunan Zheng","doi":"10.1016/j.topol.2025.109281","DOIUrl":"10.1016/j.topol.2025.109281","url":null,"abstract":"<div><div>With the background of diffeological vector spaces endowed with the D-topology, in the paper (Z. Yang and Z. Hu, 2024 <span><span>[14]</span></span>), the authors defined the concept of quasitopological vector space, which is a vector space with a topology satisfying the conditions that the vector addition is separately continuous and the scalar multiplication is continuous. Based on this, in the papers (Z. Yang and Y. Fang, 2024 <span><span>[13]</span></span>) and (Z. Yang and Q. Zheng, 2024 <span><span>[15]</span></span>), the authors continuously discussed this concept. In the present paper, we characterize the quasitopological vector space using the neighborhood system at 0, give the coproducts in the category of quasitopological vector spaces, and the free quasitopological vector space over any topological space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109281"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480601","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":"Generalizing β- and λ-maps","authors":"Ana Belén Avilez","doi":"10.1016/j.topol.2025.109282","DOIUrl":"10.1016/j.topol.2025.109282","url":null,"abstract":"<div><div>We generalize the notions of <em>β</em>- and <em>λ</em>-maps in terms of selections of sublocales, obtaining different classes of localic maps. These new classes of maps are used to characterize almost normal, extremally disconnected, <em>F</em>- and <em>Oz</em>-locales, among other types of locales, in a manner akin to the characterization of normal locales via <em>β</em>-maps. As a byproduct we obtain a characterization of localic maps that preserve the completely below relation (that is, the right adjoints of assertive frame homomorphisms).</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109282"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480604","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}

{"title":"The mod-2 cohomology groups of low-dimensional unordered flag manifolds and Auerbach bases","authors":"Lorenzo Guerra,&nbsp;Santanil Jana,&nbsp;Arun Maiti","doi":"10.1016/j.topol.2025.109279","DOIUrl":"10.1016/j.topol.2025.109279","url":null,"abstract":"<div><div>Unordered flag manifolds are the manifolds of unordered <em>n</em>-tuple of mutually orthogonal lines in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. In this paper, we develop some basic tools to compute the mod-2 cohomology groups of these spaces and apply them for explicit computation for small <em>n</em>. We show that this computation improves the known estimate of the number of Auerbach bases of normed linear spaces of small dimensions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109279"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487492","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":"Almost complex structures on sphere bundles","authors":"G.V. Ambika,&nbsp;B. Subhash","doi":"10.1016/j.topol.2025.109278","DOIUrl":"10.1016/j.topol.2025.109278","url":null,"abstract":"<div><div>In this article, we study the existence of almost complex structures on manifolds that arise as total space of sphere bundles over complex projective spaces and over closed, simply connected 4-manifolds.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109278"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480600","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}