Topology and its Applications最新文献

{"title":"Extreme Reidemeister spectra of finite groups","authors":"Sam Tertooy","doi":"10.1016/j.topol.2024.109089","DOIUrl":"10.1016/j.topol.2024.109089","url":null,"abstract":"<div><div>We extend the notions of “<span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-property” and “full (extended) Reidemeister spectrum” to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small order as well as (quasi)simple groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109089"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099705","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-affine n-valued maps on tori","authors":"K. Dekimpe ,&nbsp;L. De Weerdt","doi":"10.1016/j.topol.2024.109087","DOIUrl":"10.1016/j.topol.2024.109087","url":null,"abstract":"<div><div>In this paper we construct <em>n</em>-valued maps on <em>k</em>-dimensional tori, where <span><math><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, that are not homotopic to affine <em>n</em>-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109087"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100006","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 Borsuk–Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles","authors":"Daciberg Lima Gonçalves ,&nbsp;Vinicius Casteluber Laass ,&nbsp;Weslem Liberato Silva","doi":"10.1016/j.topol.2024.109081","DOIUrl":"10.1016/j.topol.2024.109081","url":null,"abstract":"<div><div>Let <em>M</em> and <em>N</em> be fiber bundles over the same base <em>B</em>, where <em>M</em> is endowed with a free involution <em>τ</em> over <em>B</em>. A homotopy class <span><math><mi>δ</mi><mo>∈</mo><msub><mrow><mo>[</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo>]</mo></mrow><mrow><mi>B</mi></mrow></msub></math></span> (over <em>B</em>) is said to have the Borsuk–Ulam property with respect to <em>τ</em> if for every fiber-preserving map <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> over <em>B</em> which represents <em>δ</em> there exists a point <span><math><mi>x</mi><mo>∈</mo><mi>M</mi></math></span> such that <span><math><mi>f</mi><mo>(</mo><mi>τ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. In the cases that <em>B</em> is a <span><math><mi>K</mi><mo>(</mo><mi>π</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-space and the fibers of the projections <span><math><mi>M</mi><mo>→</mo><mi>B</mi></math></span> and <span><math><mi>N</mi><mo>→</mo><mi>B</mi></math></span> are <span><math><mi>K</mi><mo>(</mo><mi>π</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> closed surfaces <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>M</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span>, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span> over <em>B</em> has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of <em>M</em>, the orbit space of <em>M</em> by <em>τ</em> and a type of generalized braid groups of <em>N</em> that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> that satisfy the Borsuk-Ulam property, with respect to all involutions <em>τ</em> over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, for the torus bundles over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> with <span><math><mi>M</mi><mo>=</mo><mi>N</mi><mo>=</mo><mi>M</mi><mi>A</mi></math></span> and <span><math><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mn>1</mn><mspace></mspace></mtd><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109081"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099703","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":"Coset spaces with certain q-sequences","authors":"Jiewen Chen,&nbsp;Bin Zhao","doi":"10.1016/j.topol.2024.109192","DOIUrl":"10.1016/j.topol.2024.109192","url":null,"abstract":"<div><div>Coset spaces with certain <em>q</em>-sequences are investigated. Firstly, we give some characterizations of strict <em>q</em>-spaces and strong <em>q</em>-spaces in coset spaces, and show that if <em>H</em> is a closed neutral subgroup of a topological group <em>G</em>, then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is a strict (resp., strong) <em>q</em>-space if and only if <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is an open sequential-perfect (resp., strongly sequential-perfect) preimage of a metrizable space. Secondly, we discuss metrizability of coset spaces, and prove that if <em>H</em> is a closed neutral subgroup of a topological group <em>G</em>, then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is metrizable if and only if <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is a sequential <em>csf</em>-countable <em>q</em>-space. Furthermore, if <em>H</em> is a compact subgroup of a topological group <em>G</em>, then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is metrizable if and only if <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is a hereditarily normal strict <em>q</em>-space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109192"},"PeriodicalIF":0.6,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147420","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":"Ultrafilters and the Katětov order","authors":"Krzysztof Kowitz,&nbsp;Adam Kwela","doi":"10.1016/j.topol.2024.109191","DOIUrl":"10.1016/j.topol.2024.109191","url":null,"abstract":"<div><div>Let <span><math><mi>I</mi></math></span> be an ideal on <em>ω</em>. Following Baumgartner (1995) <span><span>[2]</span></span>, we say that an ultrafilter <span><math><mi>U</mi></math></span> on <em>ω</em> is an <span><math><mi>I</mi></math></span>-ultrafilter if for every function <span><math><mi>f</mi><mo>:</mo><mi>ω</mi><mo>→</mo><mi>ω</mi></math></span> there is <span><math><mi>A</mi><mo>∈</mo><mi>U</mi></math></span> with <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo><mo>∈</mo><mi>I</mi></math></span>. In particular, P-points are exactly <span><math><mrow><mi>Fin</mi></mrow><mo>×</mo><mrow><mi>Fin</mi></mrow></math></span>-ultrafilters.</div><div>If there is an <span><math><mi>I</mi></math></span>-ultrafilter which is not a <span><math><mi>J</mi></math></span>-ultrafilter, then <span><math><mi>I</mi></math></span> is not below <span><math><mi>J</mi></math></span> in the Katětov order <span><math><msub><mrow><mo>⩽</mo></mrow><mrow><mi>K</mi></mrow></msub></math></span> (i.e., for every function <span><math><mi>f</mi><mo>:</mo><mi>ω</mi><mo>→</mo><mi>ω</mi></math></span> there is <span><math><mi>A</mi><mo>∈</mo><mi>I</mi></math></span> with <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>[</mo><mi>A</mi><mo>]</mo><mo>∉</mo><mi>J</mi></math></span>), however the reversed implication is not true (even consistently).</div><div>Recently it was shown that for all Borel ideals <span><math><mi>I</mi></math></span> we have: <span><math><mi>I</mi><msub><mrow><mo>≰</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mi>Fin</mi></mrow><mo>×</mo><mrow><mi>Fin</mi></mrow></math></span> if and only if in some forcing extension one can find an <span><math><mi>I</mi></math></span>-ultrafilter which is not a P-point (Filipów et al. (2022) <span><span>[6]</span></span>).</div><div>We show that under some combinatorial assumptions imposed on the ideal <span><math><mi>J</mi></math></span>, the classes of <span><math><mi>J</mi></math></span>-ultrafilters and <span><math><mrow><mi>Fin</mi></mrow><mo>×</mo><mi>J</mi></math></span>-ultrafilters coincide. This allows us to find some sufficient conditions on ideals to obtain the equivalence: <span><math><mi>I</mi><msub><mrow><mo>≰</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mi>Fin</mi></mrow><mo>×</mo><mi>J</mi></math></span> if and only if in some forcing extension one can find an <span><math><mi>I</mi></math></span>-ultrafilter which is not a <span><math><mi>J</mi></math></span>-ultrafilter. We provide several examples of ideals, for which the above equivalence is true, including the ideal of nowhere dense subsets of <span><math><mi>Q</mi></math></span> and the ideal of sets of asymptotic density zero.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109191"},"PeriodicalIF":0.6,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146940","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":"End-essential spanning surfaces for links in thickened surfaces","authors":"Thomas Kindred","doi":"10.1016/j.topol.2024.109187","DOIUrl":"10.1016/j.topol.2024.109187","url":null,"abstract":"<div><div>Let <em>D</em> be a cellular alternating link diagram on a closed orientable surface Σ. We prove that if <em>D</em> has no removable nugatory crossings then each checkerboard surface from <em>D</em> is <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-essential and contains no essential closed curve that is ∂-parallel in <span><math><mi>Σ</mi><mo>×</mo><mi>I</mi></math></span>. Our chief motivation comes from technical aspects of a companion paper, where we prove that Tait's flyping conjecture holds for alternating virtual links. We also describe possible applications via Turaev surfaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109187"},"PeriodicalIF":0.6,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147417","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":"Satellite operations and the loop expansion of the Kontsevich invariant of knots","authors":"Kouki Yamaguchi","doi":"10.1016/j.topol.2024.109189","DOIUrl":"10.1016/j.topol.2024.109189","url":null,"abstract":"<div><div>The Kontsevich invariant of knots has a special expansion, which is called the loop expansion. In this paper, we present the behavior of some loop parts, after some types of satellite operation; this types of satellite operation do not change the 1-loop part, the Alexander polynomial.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109189"},"PeriodicalIF":0.6,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147421","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":"Extending periodic maps from the genus 2 surface to the 3-torus","authors":"Weibiao Wang ,&nbsp;Yimu Zhang","doi":"10.1016/j.topol.2024.109186","DOIUrl":"10.1016/j.topol.2024.109186","url":null,"abstract":"<div><div>There are 22 nontrivial periodic maps on the orientable closed surface of genus 2, up to conjugacy. We determine whether they can extend to periodic maps on the 3-torus, orientation-preservingly or orientation-reversingly.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109186"},"PeriodicalIF":0.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146941","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":"Dichotomy theorem for topological semigroup actions","authors":"Zhumin Ding ,&nbsp;Yu Huang ,&nbsp;Tao Wang","doi":"10.1016/j.topol.2024.109188","DOIUrl":"10.1016/j.topol.2024.109188","url":null,"abstract":"<div><div>We introduce the notions of equi-stable points and equi-asymptotically stable points for topological semigroup actions with a regular system and explore the totally different behaviors of control sets based on these two kinds of points, which are the analogies to the well-known dichotomy theorem for topological transitive dynamical systems. Besides, two illustrative examples are given to show that the equi-asymptotically stable points are different from equi-stable points.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109188"},"PeriodicalIF":0.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146802","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":"Constrained motion spaces of robotic arms","authors":"Jack Pierce","doi":"10.1016/j.topol.2024.109184","DOIUrl":"10.1016/j.topol.2024.109184","url":null,"abstract":"<div><div>In this paper, we develop the theory of constrained motion spaces of robotic arms. We compute their homology groups in two cases: when the constraint is a horizontal line and when it is a smooth curve whose motion space is a smooth manifold. We show the computation of homology amounts to counting the collinear configurations, reducing a topological problem to a combinatorial problem. Our results rely on Morse theory, along with Walker's and Farber's work on polygonal linkages.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109184"},"PeriodicalIF":0.6,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147418","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}