{"title":"Expansive actions and the GCH","authors":"Luis Ferrari","doi":"10.1016/j.topol.2024.109190","DOIUrl":"10.1016/j.topol.2024.109190","url":null,"abstract":"<div><div>In this paper, we provide a generalization in terms of actions of the theorem 2.2 in <span><span>[7]</span></span> establishing a necessary and sufficient condition for the existence of expansive homeomorphisms on countable compact spaces. The study of the cardinalities of groups acting expansively will lead us to a relationship between expansivity and the Generalized Continuum Hypothesis.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109190"},"PeriodicalIF":0.6,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147015","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 Bowen-Franks type theorems for multivalued maps and impulsive differential equations","authors":"Jan Andres, Pavel Ludvík","doi":"10.1016/j.topol.2024.109080","DOIUrl":"10.1016/j.topol.2024.109080","url":null,"abstract":"<div><div>The aim of the present paper is to extend the well known Bowen-Franks type theorems to interval and circle multivalued maps. In this way, a positive topological entropy, including its lower estimate, can be implied by the existence of periodic orbits whose orders differ from a power of 2, or provided the topological degree of given circle maps is absolutely greater than 1. Simple applications are also given to impulsive differential equations.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109080"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156706","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":"R∞-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length","authors":"Karel Dekimpe , Maarten Lathouwers","doi":"10.1016/j.topol.2024.109084","DOIUrl":"10.1016/j.topol.2024.109084","url":null,"abstract":"<div><div>In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-property, while there are examples of such groups of Hirsch length 7 that do have the <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-property.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109084"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156707","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":"Generic classes defined by neighborhoods assignments and stars","authors":"Iván Martínez-Ruiz , 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}
Daciberg Lima Gonçalves , Bartira Maués , Daniel Vendrúscolo
{"title":"Nielsen fixed point theory for split n-valued maps on the Klein bottle","authors":"Daciberg Lima Gonçalves , Bartira Maués , Daniel Vendrúscolo","doi":"10.1016/j.topol.2024.109085","DOIUrl":"10.1016/j.topol.2024.109085","url":null,"abstract":"<div><div>In this work we begin the study of <em>n</em>-valued maps on the Klein bottle, denoted by <em>K</em>, where we focus on the split ones. We provide an explicit description of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo></math></span> (the <em>n</em>-th pure braid group of <em>K</em>) as an iterated semi-direct product of the form <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo>⋊</mo></mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mo>⋯</mo><mo>⋊</mo><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mo>⋊</mo></mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mo>)</mo><mo>)</mo></math></span>, where the <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>s</mi></math></span> are free groups on <em>i</em> letters. Given a split <em>n</em>-valued map <span><math><mi>Φ</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> its pointed homotopy class is determined by a pair of braids in <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. We also provide a formula for <span><math><mi>N</mi><mo>(</mo><mi>Φ</mi><mo>)</mo></math></span>, the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions <span><math><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>s</mi></math></span>. If <span><math><mi>Φ</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></math></span> is a 2-valued map with <span><math><mi>N</mi><mo>(</mo><mi>Φ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, we show that there exists at least one 2-valued map <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mo>{</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>}</mo></math></span>, such that <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is fixed point free and for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span> it holds that <span><math><mo>[</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo><mo>=</mo><mo>[</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>]</mo></math></span>,","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 , A.K.M. Libardi , D. Vendrúscolo , 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 , 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}
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