{"title":"Free circle actions on certain simply connected 7-manifolds","authors":"Fupeng Xu","doi":"10.1016/j.topol.2025.109548","DOIUrl":"10.1016/j.topol.2025.109548","url":null,"abstract":"<div><div>In this paper, we determine for which nonnegative integers <em>k</em>, <em>l</em> and for which homotopy 7-sphere Σ the manifold <span><math><mrow><mo>(</mo><mi>k</mi><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>)</mo></mrow><mi>#</mi><mrow><mo>(</mo><mi>l</mi><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></mrow><mi>#</mi><mi>Σ</mi></math></span> admits a free smooth circle action.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109548"},"PeriodicalIF":0.5,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144889480","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":"Complete metrizability of hyperspaces","authors":"László Zsilinszky","doi":"10.1016/j.topol.2025.109546","DOIUrl":"10.1016/j.topol.2025.109546","url":null,"abstract":"<div><div>Complete metrizability of the topology of bornological convergence on the hyperspace <span><math><mi>CL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of the nonempty closed subsets of a metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is characterized. As a byproduct, it is shown under (CH), that the Attouch-Wets topology on <span><math><mi>CL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is completely metrizable if and only if the Hausdorff metric topology is, which is equivalent to <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> being completely metrizable with a separable completion remainder <span><math><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>∖</mo><mi>X</mi><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109546"},"PeriodicalIF":0.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867208","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}
Pedro F. dos Santos , Carlos Florentino , Javier Orts
{"title":"Characterizing maximal varieties via Bredon cohomology","authors":"Pedro F. dos Santos , Carlos Florentino , Javier Orts","doi":"10.1016/j.topol.2025.109544","DOIUrl":"10.1016/j.topol.2025.109544","url":null,"abstract":"<div><div>We obtain a characterization of Maximal and Galois-Maximal <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-spaces (including real algebraic varieties) in terms of <span><math><mrow><mi>RO</mi></mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-graded cohomology with coefficients in the constant Mackey functor <span><math><msub><mrow><munder><mrow><mi>F</mi></mrow><mo>_</mo></munder></mrow><mrow><mn>2</mn></mrow></msub></math></span>, using the structure theorem of Clover May. Other known characterizations, for instance in terms of equivariant Borel cohomology, are also rederived from this. For the particular case of a smooth projective real variety <em>V</em>, equivariant Poincaré duality is used to deduce further symmetry restrictions for the decomposition of the <span><math><mrow><mi>RO</mi></mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-graded cohomology of the complex locus <span><math><mi>V</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> given by the same structure theorem. We illustrate this result with some computations, including the <span><math><mrow><mi>RO</mi></mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-graded cohomology with <span><math><msub><mrow><munder><mrow><mi>F</mi></mrow><mo>_</mo></munder></mrow><mrow><mn>2</mn></mrow></msub></math></span> coefficients of real <em>K</em>3 surfaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109544"},"PeriodicalIF":0.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902632","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":"Intermediate algebras of semialgebraic functions","authors":"E. Baro, José F. Fernando, J.M. Gamboa","doi":"10.1016/j.topol.2025.109547","DOIUrl":"10.1016/j.topol.2025.109547","url":null,"abstract":"<div><div>We characterize intermediate <span><math><mi>R</mi></math></span>-algebras <em>A</em> between the ring of semialgebraic functions <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and the ring <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of bounded semialgebraic functions on a semialgebraic set <em>X</em> as rings of fractions of <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. This allows us to compute the Krull dimension of <em>A</em>, the transcendence degree over <span><math><mi>R</mi></math></span> of the residue fields of <em>A</em> and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean <span><math><mi>R</mi></math></span>-algebras <em>A</em>. In addition we study intermediate <span><math><mi>R</mi></math></span>-algebras generated by proper ideals and we prove an extension theorem for functions in such <span><math><mi>R</mi></math></span>-algebras.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109547"},"PeriodicalIF":0.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867207","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}
David Bryant , Katharina T. Huber , Vincent Moulton , Andreas Spillner
{"title":"Subtree distances, tight spans and diversities","authors":"David Bryant , Katharina T. Huber , Vincent Moulton , Andreas Spillner","doi":"10.1016/j.topol.2025.109545","DOIUrl":"10.1016/j.topol.2025.109545","url":null,"abstract":"<div><div>We characterize when a set of distances <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> between elements in a set <em>X</em> have a <em>subtree representation</em>, a real tree <em>T</em> and a collection <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub></math></span> of subtrees of <em>T</em> such that <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> equals the length of the shortest path in <em>T</em> from a point in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> to a point in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. The characterization was first established for <em>finite X</em> by Hirai (2006) using a tight span construction defined for <em>distance spaces</em>, metric spaces without the triangle inequality. To extend Hirai's result beyond finite <em>X</em> we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity – a generalization of a metric space which assigns values to all finite subsets of <em>X</em>, not just to pairs – has a tight span which is tree-like.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109545"},"PeriodicalIF":0.5,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867160","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":"Two observations on extremally disconnected topological groups","authors":"Yi Zhou , Jialiang He , Hang Zhang , Shuguo Zhang","doi":"10.1016/j.topol.2025.109543","DOIUrl":"10.1016/j.topol.2025.109543","url":null,"abstract":"<div><div>By modifying a method of Malykhin's, we construct two Hausdorff group topologies on the uncountable Boolean group <span><math><mo>(</mo><msup><mrow><mo>[</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow><mrow><mo><</mo><mi>ω</mi></mrow></msup><mo>,</mo><mo>▵</mo><mo>)</mo></math></span> which are both nondiscrete and extremally disconnected. This is accomplished by working under ZFC plus Jensen's Diamond Principle. The first one has the property that all subgroups of the form <span><math><msup><mrow><mo>[</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∖</mo><mi>α</mi><mo>]</mo></mrow><mrow><mo><</mo><mi>ω</mi></mrow></msup></math></span> are dense and all countable subsets of <span><math><msup><mrow><mo>[</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow><mrow><mo><</mo><mi>ω</mi></mrow></msup></math></span> are closed and discrete. This answers a question posed by C.A. Martínez-Ranero and U.A. Ramos-García <span><span>[7, Question 3.4]</span></span>. The second one has the property that some subgroup (endowed with the subspace topology) fails to be extremally disconnected. This answers a question posed by Arhangel'skii and Tkachenko <span><span>[2, Open Problems 4.5.1]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109543"},"PeriodicalIF":0.5,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860875","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}
Claudio Agostini, Andrea Medini, Lyubomyr Zdomskyy
{"title":"Countable dense homogeneity and topological groups","authors":"Claudio Agostini, Andrea Medini, Lyubomyr Zdomskyy","doi":"10.1016/j.topol.2025.109537","DOIUrl":"10.1016/j.topol.2025.109537","url":null,"abstract":"<div><div>Building on results of Medvedev, we construct a <span><math><mi>ZFC</mi></math></span> example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> of size <span><math><mi>b</mi></math></span> that is a <em>λ</em>-set. We also conjecture that every countable dense homogeneous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109537"},"PeriodicalIF":0.5,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860871","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":"M⁎, N⁎, and H⁎","authors":"Will Brian , Alan Dow , Klaas Pieter Hart","doi":"10.1016/j.topol.2025.109539","DOIUrl":"10.1016/j.topol.2025.109539","url":null,"abstract":"<div><div>Let <span><math><mi>M</mi><mo>=</mo><mi>N</mi><mo>×</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. The natural projection <span><math><mi>π</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span>, which sends <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> to <em>n</em>, induces a projection mapping <span><math><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, where <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> denote the Čech-Stone remainders of <span><math><mi>M</mi></math></span> and <span><math><mi>N</mi></math></span>, respectively.</div><div>We show that <span><math><mi>CH</mi></math></span> implies every autohomeomorphism of <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> lifts through the natural projection to an autohomeomorphism of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. That is, for every homeomorphism <span><math><mi>h</mi><mo>:</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> there is a homeomorphism <span><math><mi>H</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that <span><math><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∘</mo><mi>H</mi><mo>=</mo><mi>h</mi><mo>∘</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. This complements a recent result of the second author, who showed that this lifting property is not a consequence of <span><math><mi>ZFC</mi></math></span>.</div><div>Combining this lifting theorem with a recent result of the first author, we also prove that <span><math><mi>CH</mi></math></span> implies there is an order-reversing autohomeomorphism of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the Čech-Stone remainder of the half line <span><math><mi>H</mi><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109539"},"PeriodicalIF":0.5,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810505","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":"Classification of minimal separating sets of low genus surfaces","authors":"Christopher N. Aagaard, J.J.P. Veerman","doi":"10.1016/j.topol.2025.109540","DOIUrl":"10.1016/j.topol.2025.109540","url":null,"abstract":"<div><div>A minimal separating set in a connected topological space <em>X</em> is a subset <span><math><mi>L</mi><mo>⊂</mo><mi>X</mi></math></span> with the property that <span><math><mi>X</mi><mo>∖</mo><mi>L</mi></math></span> is disconnected, but if <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is a proper subset of <em>L</em>, then <span><math><mi>X</mi><mo>∖</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is connected. Such sets appear in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points <em>p</em> and <em>q</em>, then the set of points <em>x</em> satisfying <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> are minimal separating. Here we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graph embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and underlying graphs.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109540"},"PeriodicalIF":0.5,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830359","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":"Residual finiteness and profinite properties of automorphism groups of quandles","authors":"Manpreet Singh","doi":"10.1016/j.topol.2025.109538","DOIUrl":"10.1016/j.topol.2025.109538","url":null,"abstract":"<div><div>We explore residual finiteness and profinite properties of automorphism groups of quandles. We prove that the automorphism group of finitely generated residually finite quandles is residually finite. We establish a similar result for a broader class of quandles. As an application, we prove that the automorphism group of the fundamental quandle of a link in the 3-sphere is residually finite. Furthermore, we prove that the welded braid group on <em>n</em> strands is residually finite. We provide a topological characterization of profinite quandles. Moreover, for finitely generated profinite quandles, we prove that the automorphism groups and the inner automorphism groups are profinite.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109538"},"PeriodicalIF":0.5,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810504","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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