Topology and its Applications最新文献

{"title":"Various notions of shadowing in triangular system and its component systems","authors":"Deepanshu Dhawan,&nbsp;Puneet Sharma","doi":"10.1016/j.topol.2024.109109","DOIUrl":"10.1016/j.topol.2024.109109","url":null,"abstract":"<div><div>In this article, we investigate various forms of shadowing for a general triangular system. In particular, we relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. We prove that if the base <em>f</em> for <em>T</em> is transitive then shadowing in the base map and the non-autonomous system generated by a transitive point ensures shadowing of the triangular system. We prove that if the base map for <em>T</em> is expansive then shadowing in the triangular system ensures shadowing in the component systems. We prove that if the non-autonomous component systems form a synchronized family and the base map possesses a globally attracting fixed point <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> then eventual shadowing in system generated by <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> ensures eventual shadowing in each of the non-autonomous component systems. We also investigate chain transitivity and chain mixing for a general triangular system.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109109"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534568","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":"Non-abelian tensor product and circular orderability of groups","authors":"Maxim Ivanov","doi":"10.1016/j.topol.2024.109111","DOIUrl":"10.1016/j.topol.2024.109111","url":null,"abstract":"<div><div>For a group <em>G</em> we consider its tensor square <span><math><mi>G</mi><mo>⊗</mo><mi>G</mi></math></span> and exterior square <span><math><mi>G</mi><mo>∧</mo><mi>G</mi></math></span>. We prove that for a circularly orderable group <em>G</em>, under some assumptions on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group <em>G</em> to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109111"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552383","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":"An upper bound for the number of critical points of the systole function on the moduli space of hyperbolic surfaces","authors":"Yue Gao","doi":"10.1016/j.topol.2024.109091","DOIUrl":"10.1016/j.topol.2024.109091","url":null,"abstract":"<div><div>We obtain an upper bound for the number of critical points of the systole function on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. Besides, we obtain an upper bound for the number of those critical points whose systole is smaller than a constant.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109091"},"PeriodicalIF":0.6,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530753","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":"Degenerations of the product geometries in projective space that contain Nil","authors":"Thomas Shifley ,&nbsp;Steve Trettel","doi":"10.1016/j.topol.2024.109078","DOIUrl":"10.1016/j.topol.2024.109078","url":null,"abstract":"<div><div>This paper produces explicit conjugacy paths for the product geometries <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span> and <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span> whose limits contain the geometry of the Heisenberg group's action on itself. These are the first such conjugacy limits to any model of Nil, continuing the program of Daryl Cooper, Jeffrey Danciger, and Anna Wienhard to determine all possible degenerations between Thurston geometries in <span><math><mo>(</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>4</mn><mo>,</mo><mi>R</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>RP</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109078"},"PeriodicalIF":0.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445193","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":"Hattori subspaces","authors":"Angel Calderón-Villalobos ,&nbsp;Iván Sánchez","doi":"10.1016/j.topol.2024.109077","DOIUrl":"10.1016/j.topol.2024.109077","url":null,"abstract":"<div><div>For a subset <em>A</em> of an almost topological group <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, the Hattori space <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a topological space whose underlying set is <em>G</em> and whose topology <span><math><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is defined as follows: if <span><math><mi>x</mi><mo>∈</mo><mi>A</mi></math></span> (respectively, <span><math><mi>x</mi><mo>∉</mo><mi>A</mi></math></span>), then the neighborhoods of <em>x</em> in <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> are the same neighborhoods of <em>x</em> in the reflection group <span><math><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></math></span> (respectively, <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>). Given an infinite subset <em>X</em> of an almost topological group <em>G</em> and <span><math><mi>A</mi><mo>⊆</mo><mi>X</mi></math></span>, we denote by <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> and <em>X</em> to the spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, respectively. We say that <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the Hattori subspace associated to <em>A</em>. In this paper, we obtain information about Hattori subspaces. We show that some known topological spaces can be obtained as Hattori subspaces of some almost topological groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109077"},"PeriodicalIF":0.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534567","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":"Velichko's notions close to sequential separability and their hereditary variants in Cp-theory","authors":"Alexander V. Osipov","doi":"10.1016/j.topol.2024.109076","DOIUrl":"10.1016/j.topol.2024.109076","url":null,"abstract":"<div><div>A space <em>X</em> is <em>sequentially separable</em> if there is a countable <span><math><mi>S</mi><mo>⊂</mo><mi>X</mi></math></span> such that every point of <em>X</em> is the limit of a sequence of points from <em>S</em>. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: <em>σ-separability</em> and <em>F-separability</em>. The aim of this paper is to study <em>σ</em>-separability and <em>F</em>-separability (and their hereditary variants) of the space <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of all real-valued continuous functions, defined on a Tychonoff space <em>X</em>, endowed with the pointwise convergence topology. In particular, we proved that <em>σ</em>-separability coincides with sequential separability. Hereditary variants (hereditarily <em>σ</em>-separability and hereditarily <em>F</em>-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109076"},"PeriodicalIF":0.6,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417072","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 complete invariant for shift equivalence for Boolean matrices and finite relations","authors":"Ethan Akin ,&nbsp;Marian Mrozek ,&nbsp;Mateusz Przybylski ,&nbsp;Jim Wiseman","doi":"10.1016/j.topol.2024.109075","DOIUrl":"10.1016/j.topol.2024.109075","url":null,"abstract":"<div><div>We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those components.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109075"},"PeriodicalIF":0.6,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417137","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 diagonal degrees and star networks","authors":"Nathan Carlson","doi":"10.1016/j.topol.2024.109074","DOIUrl":"10.1016/j.topol.2024.109074","url":null,"abstract":"<div><div>Given an open cover <span><math><mi>U</mi></math></span> of a topological space <em>X</em>, we introduce the notion of a star network for <span><math><mi>U</mi></math></span>. The associated cardinal function <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, where <span><math><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, is used to establish new cardinal inequalities involving diagonal degrees. We show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>s</mi><mi>n</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> space <em>X</em>, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. One result has as corollaries Buzyakova's theorem that a ccc space with a regular <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-diagonal has cardinality at most <span><math><mi>c</mi></math></span>, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the property <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>a</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span> and use the Erdős-Rado theorem to show that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mover><mrow><mi>Δ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Urysohn space <em>X</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109074"},"PeriodicalIF":0.6,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417071","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":"Approximations by disjoint subcontinua and indecomposability","authors":"David S. Lipham","doi":"10.1016/j.topol.2024.109071","DOIUrl":"10.1016/j.topol.2024.109071","url":null,"abstract":"<div><div>We study approximations of continuum-wise connected spaces, or <em>semicontinua</em>, and show that every indecomposable semicontinuum can be approximated from within by a sequence of pairwise disjoint continua. As a corollary, we find that if <em>X</em> is a <em>G</em>-like continuum or a one-dimensional non-separating plane continuum, which is the closure of an indecomposable semicontinuum, then <em>X</em> is indecomposable. We also prove that a composant of an indecomposable continuum cannot be embedded into a Suslinian continuum.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109071"},"PeriodicalIF":0.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417070","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":"Calculation of Nielsen periodic numbers on infra-solvmanifolds","authors":"Changbok Li","doi":"10.1016/j.topol.2024.109073","DOIUrl":"10.1016/j.topol.2024.109073","url":null,"abstract":"<div><div>Recently, a formula for computing the Nielsen periodic numbers <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of self maps <em>f</em> on infra-nilmanifolds and infra-solvmanifolds of type (R) was found. In this paper, we extend this formula to the case of general infra-solvmanifolds. We show that infra-solvmanifolds are essentially reducible to the GCD and essentially toral, and determine conditions under which <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. We show that the prime Nielsen-Jiang periodic number <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of a self map <em>f</em> on an infra-solvmanifold <em>M</em> can be calculated by Nielsen numbers of lifts of suitable iterates of <em>f</em> to an <span><math><mi>NR</mi></math></span>-solvmanifold that finitely covers <em>M</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109073"},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326718","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}