Topology and its Applications最新文献

{"title":"Generic classes defined by neighborhoods assignments and stars","authors":"Iván Martínez-Ruiz ,&nbsp;Alejandro Ramírez-Páramo","doi":"10.1016/j.topol.2024.109147","DOIUrl":"10.1016/j.topol.2024.109147","url":null,"abstract":"<div><div>In this article we have a couple of mains. The first one is to make a generic exposure on the classes that are obtained by using neighborhood assignments or applying the star operator to a given topological property <span><math><mi>P</mi></math></span>. The second one is to perform an analysis on the classes defined in the first part of the work, for the numerability property.</div><div>It is important to note that in <span><span>Example 3.13</span></span> we present a Hausdorff weakly star countable space which is not feebly Lindelof, which gives a negative response to the Question 3.14 from <span><span>[1]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109147"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164904","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 note on the existence of the Reidemeister zeta function on groups","authors":"Jonas Deré","doi":"10.1016/j.topol.2024.109088","DOIUrl":"10.1016/j.topol.2024.109088","url":null,"abstract":"<div><div>Given an endomorphism <span><math><mi>φ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math></span> on a group <em>G</em>, one can define the Reidemeister number <span><math><mi>R</mi><mo>(</mo><mi>φ</mi><mo>)</mo><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span>, by using the Reidemeister numbers <span><math><mi>R</mi><mo>(</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> of iterates <span><math><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated.</div><div>However, in order to define the Reidemeister zeta function of an endomorphism <em>φ</em>, it is necessary that the Reidemeister numbers <span><math><mi>R</mi><mo>(</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> of all iterates <span><math><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> are finite. This puts restrictions, not only on the endomorphism <em>φ</em>, but also on the possible groups <em>G</em> if <em>φ</em> is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism <em>φ</em>, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109088"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099704","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":"Nielsen fixed point theory for split n-valued maps on the Klein bottle","authors":"Daciberg Lima Gonçalves ,&nbsp;Bartira Maués ,&nbsp;Daniel Vendrúscolo","doi":"10.1016/j.topol.2024.109085","DOIUrl":"10.1016/j.topol.2024.109085","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this work we begin the study of &lt;em&gt;n&lt;/em&gt;-valued maps on the Klein bottle, denoted by &lt;em&gt;K&lt;/em&gt;, where we focus on the split ones. We provide an explicit description of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; (the &lt;em&gt;n&lt;/em&gt;-th pure braid group of &lt;em&gt;K&lt;/em&gt;) as an iterated semi-direct product of the form &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⋊&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;⋊&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⋊&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where the &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are free groups on &lt;em&gt;i&lt;/em&gt; letters. Given a split &lt;em&gt;n&lt;/em&gt;-valued map &lt;span&gt;&lt;math&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; its pointed homotopy class is determined by a pair of braids in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We also provide a formula for &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. If &lt;span&gt;&lt;math&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a 2-valued map with &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we show that there exists at least one 2-valued map &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, such that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is fixed point free and for &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; it holds that &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;,","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109085"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100022","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":"Computing the one-parameter Nielsen number for homotopies on the n-torus","authors":"Weslem Liberato Silva","doi":"10.1016/j.topol.2024.109082","DOIUrl":"10.1016/j.topol.2024.109082","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mi>I</mi><mo>→</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> be a homotopy on a n-dimensional torus. The main purpose of this paper is to present a formula for the one-parameter Nielsen number <span><math><mi>N</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> of <em>F</em> in terms of its induced homomorphism. If <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is the one-parameter Lefschetz class of <em>F</em> then <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is given by <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mspace></mspace><mi>N</mi><mo>(</mo><mi>F</mi><mo>)</mo><mi>α</mi></math></span>, for some <span><math><mi>α</mi><mo>∈</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>,</mo><mi>Z</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109082"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156651","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 minimal fixed points set of fiber preserving maps of S1-bundles over S1","authors":"D.L. Gonçalves ,&nbsp;A.K.M. Libardi ,&nbsp;D. Vendrúscolo ,&nbsp;J.P. Vieira","doi":"10.1016/j.topol.2024.109083","DOIUrl":"10.1016/j.topol.2024.109083","url":null,"abstract":"<div><div>In this work, we describe the minimal fixed points set of fiberwise maps of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundles over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, up to fiberwise homotopies. There are two fibrations under these conditions, one orientable, the torus, and the other non-orientable, the Klein bottle. For fiberwise maps the minimal fixed points set can be empty, otherwise it is described as the finite union of disjoint circles. We present models for which the fixed points set are minimal, where minimal means that no proper subset can be realized as the fixed points set in the same fiberwise homotopy class.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109083"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099702","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":"Special Issue - ‘Nielsen theory and related topics: a special issue dedicated to the memory of Robert F. Brown’","authors":"","doi":"10.1016/S0166-8641(24)00349-3","DOIUrl":"10.1016/S0166-8641(24)00349-3","url":null,"abstract":"","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109164"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100053","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":"Fixed point index bounds for self-maps on surfacelike complexes","authors":"D.L. Gonçalves ,&nbsp;M.R. Kelly","doi":"10.1016/j.topol.2024.109086","DOIUrl":"10.1016/j.topol.2024.109086","url":null,"abstract":"<div><div>For a certain family of aspherical 2-complexes it is shown that a pair of inequalities, known as hyperbolic index bounds, involving fixed point indices are satisfied for all fixed point minimal self-maps. As a corollary we verify the hyperbolic index bounds for the Nielsen fixed point classes of self-maps <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span>, when <em>X</em> is a finite wedge of compact surfaces each having non-positive Euler characteristic.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109086"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100023","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":"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}