{"title":"Cohomology bases of toric surfaces","authors":"Xin Fu , Tseleung So , Jongbaek Song","doi":"10.1016/j.topol.2025.109392","DOIUrl":"10.1016/j.topol.2025.109392","url":null,"abstract":"<div><div>Given a compact toric surface, the multiplication of its rational cohomology can be described in terms of the intersection products of Weil divisors, or in terms of the cup products of cohomology classes representing specific cells. In this paper, we aim to compare these two descriptions. More precisely, we define two different cohomology bases, the <em>Poincaré dual basis</em> and the <em>cellular basis</em>, which give rise to matrices representing the intersection product and the cup product. We prove that these representing matrices are inverse of each other.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109392"},"PeriodicalIF":0.6,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863528","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":"Topological properties determined by closures of countable sets","authors":"V.V. Tkachuk","doi":"10.1016/j.topol.2025.109393","DOIUrl":"10.1016/j.topol.2025.109393","url":null,"abstract":"<div><div>We establish that a locally convex space must be metrizable if it has a Čech-complete dense subspace and show that there are locally convex spaces <em>L</em> of arbitrarily large extent such that <span><math><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></math></span> is <em>σ</em>-compact for any countable set <span><math><mi>A</mi><mo>⊂</mo><mi>L</mi></math></span>. Under Jensen's Axiom (⋄), we give an example of a metrizable space <em>M</em> which does not have a dense Čech-complete subspace while <span><math><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></math></span> is Čech-complete for every countable set <span><math><mi>A</mi><mo>⊂</mo><mi>M</mi></math></span>. Our results give a consistent answer to two published open questions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109393"},"PeriodicalIF":0.6,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859153","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":"On isometric universality of spaces of metrics","authors":"Yoshito Ishiki , Katsuhisa Koshino","doi":"10.1016/j.topol.2025.109394","DOIUrl":"10.1016/j.topol.2025.109394","url":null,"abstract":"<div><div>A metric space <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>. In this paper, for a metrizable space <em>Z</em> possessing abundant subspaces, we first investigate the universality of the space of metrics on <em>Z</em>. Next, in contrast, we show that if <em>Z</em> is an infinite discrete space, then the space of metrics on <em>Z</em> is universal for all metric spaces having the same weight of <em>Z</em>. As a corollary of our results, if <em>Z</em> is non-compact, or uncountable and compact, then the space of metrics on <em>Z</em> is universal for all compact metric spaces. In addition, if <em>Z</em> is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on <em>Z</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109394"},"PeriodicalIF":0.6,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863527","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":"Bar-Natan theory and tunneling between incompressible surfaces in 3-manifolds","authors":"Uwe Kaiser","doi":"10.1016/j.topol.2025.109390","DOIUrl":"10.1016/j.topol.2025.109390","url":null,"abstract":"<div><div>In <span><span>[16]</span></span> the author defined for each commutative Frobenius algebra a skein module of surfaces in a 3-manifold <em>M</em> bounding a closed 1-manifold <span><math><mi>α</mi><mo>⊂</mo><mo>∂</mo><mi>M</mi></math></span>. The surface components are colored by elements of the Frobenius algebra. The modules are called the Bar-Natan modules of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span>. In this article we show that Bar-Natan modules are colimit modules of functors associated to Frobenius algebras, <em>decoupling</em> topology from algebra. The functors are defined on a category of 3-dimensional compression bordisms embedded in cylinders over <em>M</em> and take values in a linear category defined from the Frobenius algebra. The relation with the <span><math><mn>1</mn><mo>+</mo><mn>1</mn></math></span>-dimensional topological quantum field theory functor associated to the Frobenius algebra is studied. We show that the geometric content of the skein modules is contained in a <em>tunneling graph</em> of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span>, providing a natural presentation of the Bar-Natan module by application of the functor defined from the algebra. Such presentations have essentially been stated in <span><span>[16]</span></span> and <span><span>[2]</span></span> using ad-hoc arguments. But they appear naturally on the background of the Bar-Natan functor and associated categorical considerations. We discuss in general how to deduce presentations of colimit modules for functors into module categories in terms of minimal terminal sets of objects of the category in the categorical setting. We also sketch the construction of a bicategory version of the Bar-Natan functor.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109390"},"PeriodicalIF":0.6,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863529","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}
Enrique Castañeda-Alvarado, David Maya, Miguel Angel Morales-Bautista, Fernando Orozco-Zitli
{"title":"Induced mappings on Pixley-Roy hyperspace","authors":"Enrique Castañeda-Alvarado, David Maya, Miguel Angel Morales-Bautista, Fernando Orozco-Zitli","doi":"10.1016/j.topol.2025.109389","DOIUrl":"10.1016/j.topol.2025.109389","url":null,"abstract":"<div><div>The symbol <span><math><mrow><mi>PR</mi></mrow><mo>[</mo><mi>X</mi><mo>]</mo></math></span> denotes the hyperspace consisting of all nonempty finite subsets of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> topological space <em>X</em> endowed with the Pixley-Roy topology. For an onto mapping between <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> spaces <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span>, define the induced mapping <span><math><mrow><mi>PR</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><mrow><mi>PR</mi></mrow><mo>[</mo><mi>X</mi><mo>]</mo><mo>→</mo><mrow><mi>PR</mi></mrow><mo>[</mo><mi>Y</mi><mo>]</mo></math></span> by <span><math><mrow><mi>PR</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>:</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>}</mo></math></span> (the image of <em>A</em> under <em>f</em>). In this paper, we study the relationship between the condition <em>f</em> belongs to a class of mappings between <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> spaces <span><math><mi>M</mi></math></span> and the condition <span><math><mrow><mi>PR</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo></math></span> belongs to <span><math><mi>M</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109389"},"PeriodicalIF":0.6,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844203","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":"Normal forms for rational 3-tangles","authors":"Bo-hyun Kwon , Jung Hoon Lee","doi":"10.1016/j.topol.2025.109388","DOIUrl":"10.1016/j.topol.2025.109388","url":null,"abstract":"<div><div>A collection of properly embedded three disjoint simple arcs in <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>6</mn></mrow></msub></math></span> represents a rational 3-tangle. In this paper, we define a <em>normal form</em> of collections of three disjoint <em>bridge arcs</em> for a given rational 3-tangle. We show that there is a sequence of operations called <em>normal jump moves</em> which makes a path between arbitrary two elements in the set of normal forms of the same rational 3-tangle. We believe that the normal form would give a clue to classify rational 3-tangles.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109388"},"PeriodicalIF":0.6,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826279","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":"Differential forms on diffeological spaces and diffeological gluing, I","authors":"Ekaterina Pervova","doi":"10.1016/j.topol.2025.109387","DOIUrl":"10.1016/j.topol.2025.109387","url":null,"abstract":"<div><div>This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available, that of the vector space <span><math><msup><mrow><mi>Ω</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of all <em>m</em>-forms, and that of the pseudo-bundle <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of values of these forms. We describe the behavior of the former under a gluing of diffeological spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109387"},"PeriodicalIF":0.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815223","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}
Francisco Antonio-Balderas , Leobardo Fernández , José Jirash , Jorge Moreno-Montes , Leonel Rito
{"title":"Furstenberg type theorems for strongly transitive maps","authors":"Francisco Antonio-Balderas , Leobardo Fernández , José Jirash , Jorge Moreno-Montes , Leonel Rito","doi":"10.1016/j.topol.2025.109386","DOIUrl":"10.1016/j.topol.2025.109386","url":null,"abstract":"<div><div>Given a dynamical system <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> and a positive integer <em>n</em>, we study the concept of strong transitivity in the induced map <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>×</mo><mi>n</mi></mrow></msup><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. The main problem is to figure out when the strong transitivity of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>×</mo><mn>2</mn></mrow></msup></math></span> implies the strong transitivity of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>×</mo><mi>n</mi></mrow></msup></math></span>. In certain spaces, such as finite connected graphs, we show that if <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>×</mo><mn>2</mn></mrow></msup></math></span> is strongly transitive then <em>f</em> is strongly mixing which also implies that <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>×</mo><mi>n</mi></mrow></msup></math></span> is strongly transitive for any <em>n</em>. We also give an example of a map <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> which is strongly mixing but not exact, showing that our result is optimal in finite graphs.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109386"},"PeriodicalIF":0.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815222","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 homotopy approach for maps with a continuous inclusion property","authors":"Donal O'Regan","doi":"10.1016/j.topol.2025.109384","DOIUrl":"10.1016/j.topol.2025.109384","url":null,"abstract":"<div><div>This paper presents fixed point theorems, Leray–Schauder alternatives and homotopy theorems for compact maps which have particular selection properties.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"368 ","pages":"Article 109384"},"PeriodicalIF":0.6,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817505","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 trunk number of satellite knots and Thurston norm","authors":"Zehan Pan","doi":"10.1016/j.topol.2025.109383","DOIUrl":"10.1016/j.topol.2025.109383","url":null,"abstract":"<div><div>Assume <span><math><mi>J</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> is a non-trivial knot, and assume <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a satellite pattern. Let <em>N</em> be the generalized Thurston norm of the homology class of the meridian disk in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with respect to <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. Let <em>K</em> be the satellite knot of <em>J</em> with pattern <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. We show that the trunk number of <em>K</em> is strictly greater than <em>N</em> times the trunk number of <em>J</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"369 ","pages":"Article 109383"},"PeriodicalIF":0.6,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808523","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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