Pub Date : 2025-11-24DOI: 10.1016/j.topol.2025.109672
Khadijeh Ghasemi , Ali S. Janfada , Hadi Zare
Our aim is to study bordism of immersions beyond the metastable range, so . We focus on the case of and show that for any immersion of a nonboundary , M is bordant to and its triple point manifold is a boundary too. By contrast, we show that if is an immersion whose double point manifold is not a boundary then, up to addition in the relevant bordism group, f is bordant to an immersion of a boundary. Along the way, we record that an application of the Gaussian elimination over can be used to determine the submodule of A-annihilated elements in .”. We also review the method of computing the submodule of primitive elements in for X being path connected.
我们的目的是研究在亚稳范围以外的浸入Mn↑R2n−6的谱性，因此n∈{6,7，…，12}。我们着重讨论了n=11的情况，并证明了对于任意浸入的非边界f:M11 - R16， M是与P6×V5的边界，并且它的三点流形也是一个边界。通过对比，我们证明了如果f:M11 - R16是一个浸入，其双点流形不是边界，则在相应的边界群中，f与一个边界的浸入相邻。在此过程中，我们记录到在Z/2上应用高斯消去可以用来确定HnX中a湮灭元素的子模。我们也回顾了在H ^ H ^ QX中，当X是路径连通时，基元子模的计算方法。
{"title":"On the Gaussian elimination and bordism of certain immersions beyond the metastable range","authors":"Khadijeh Ghasemi , Ali S. Janfada , Hadi Zare","doi":"10.1016/j.topol.2025.109672","DOIUrl":"10.1016/j.topol.2025.109672","url":null,"abstract":"<div><div>Our aim is to study bordism of immersions <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>6</mn></mrow></msup></math></span> beyond the metastable range, so <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>12</mn><mo>}</mo></math></span>. We focus on the case of <span><math><mi>n</mi><mo>=</mo><mn>11</mn></math></span> and show that for any immersion of a nonboundary <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>16</mn></mrow></msup></math></span>, <em>M</em> is bordant to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>×</mo><msup><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> and its triple point manifold is a boundary too. By contrast, we show that if <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>16</mn></mrow></msup></math></span> is an immersion whose double point manifold is not a boundary then, up to addition in the relevant bordism group, <em>f</em> is bordant to an immersion of a boundary. Along the way, we record that an application of the Gaussian elimination over <span><math><mi>Z</mi><mo>/</mo><mn>2</mn></math></span> can be used to determine the submodule of A-annihilated elements in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>X</mi></math></span>.”. We also review the method of computing the submodule of primitive elements in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mi>Q</mi><mi>X</mi></math></span> for <em>X</em> being path connected.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109672"},"PeriodicalIF":0.5,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600404","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}
Pub Date : 2025-11-24DOI: 10.1016/j.topol.2025.109668
Faraz Ahmad
We study some metric and topological properties of spaces of equivariant operators. We define the notions of a compact perception pair, compactification of a perception pair, and compactification of a space of group equivariant non-expansive operators. We prove that every perception pair with totally bounded space of measurements, which is also rich enough to endow the common domain with a metric structure, can be isometrically embedded in a compact perception pair. Likewise, we prove that if the images of group equivariant non-expansive operators in a given space form a cover for their common codomain, then the space of such operators can be isometrically embedded in a compact space of group equivariant non-expansive operators, such that the new reference perception pairs are compactifications of the original ones having totally bounded data sets. Meanwhile, we state some compatibility conditions for these embeddings and show that they too are satisfied by our constructions.
{"title":"Compactification of perception pairs and spaces of group equivariant non-expansive operators","authors":"Faraz Ahmad","doi":"10.1016/j.topol.2025.109668","DOIUrl":"10.1016/j.topol.2025.109668","url":null,"abstract":"<div><div>We study some metric and topological properties of spaces of equivariant operators. We define the notions of a compact perception pair, compactification of a perception pair, and compactification of a space of group equivariant non-expansive operators. We prove that every perception pair with totally bounded space of measurements, which is also rich enough to endow the common domain with a metric structure, can be isometrically embedded in a compact perception pair. Likewise, we prove that if the images of group equivariant non-expansive operators in a given space form a cover for their common codomain, then the space of such operators can be isometrically embedded in a compact space of group equivariant non-expansive operators, such that the new reference perception pairs are compactifications of the original ones having totally bounded data sets. Meanwhile, we state some compatibility conditions for these embeddings and show that they too are satisfied by our constructions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109668"},"PeriodicalIF":0.5,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685269","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}
Pub Date : 2025-11-24DOI: 10.1016/j.topol.2025.109670
Allan Edley Ramos de Andrade , Northon Canevari Leme Penteado , Sergio Tsuyoshi Ura
Given G a finite group which acts on and H a normal cyclic subgroup of prime order, in [1] the authors have defined and estimated the cohomological dimension of the set of -coincidence points of an acyclic multi-valued map relative to an essential map , where X is a compact Hausdorff space and Y is a k-dimensional CW-complex. In this work we extended this result to an admissible multi-valued map and to an essential multi-valued map such that . Furthermore, we define and estimate the cohomological dimension of a coincidence set where , are two admissible multi-valued maps and N is a connected closed manifold and a homology n-sphere.
给定G是作用于Sn的有限群，H是素阶的正规循环子群，在[1]中定义并估计了非循环多值映射F:X X→Sn的（H，G）-重合点的集合a φ（F，H，G）相对于本质映射φ:X→Sn的上同调维数，其中X是紧Hausdorff空间，Y是k维cw复形。在本工作中，我们将此结果推广到一个容许多值映射F:X X Y和一个本质多值映射Φ：X X Sn，使得Ti（Φ(X)）∩Φ(X)=∅,i=1，…，p。进一步，我们定义并估计了重合集a （F，Φ）的上同调维数，其中F:X × M， Φ：X × N是两个可容许的多值映射，N是连通闭流形和同调N球。
{"title":"A nonsymmetric approach to coincidences of admissible multi-valued maps","authors":"Allan Edley Ramos de Andrade , Northon Canevari Leme Penteado , Sergio Tsuyoshi Ura","doi":"10.1016/j.topol.2025.109670","DOIUrl":"10.1016/j.topol.2025.109670","url":null,"abstract":"<div><div>Given <em>G</em> a finite group which acts on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and <em>H</em> a normal cyclic subgroup of prime order, in <span><span>[1]</span></span> the authors have defined and estimated the cohomological dimension of the set <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> of <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span>-coincidence points of an acyclic multi-valued map <span><math><mi>F</mi><mo>:</mo><mi>X</mi><mo>⊸</mo><mi>Y</mi></math></span> relative to an essential map <span><math><mi>φ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <em>X</em> is a compact Hausdorff space and <em>Y</em> is a <em>k</em>-dimensional CW-complex. In this work we extended this result to an admissible multi-valued map <span><math><mi>F</mi><mo>:</mo><mi>X</mi><mo>⊸</mo><mi>Y</mi></math></span> and to an essential multi-valued map <span><math><mi>Φ</mi><mo>:</mo><mi>X</mi><mo>⊸</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>Φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>∩</mo><mi>Φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∅</mo><mo>,</mo><mspace></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi></math></span>. Furthermore, we define and estimate the cohomological dimension of a coincidence set <span><math><mi>A</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>Φ</mi><mo>)</mo></math></span> where <span><math><mi>F</mi><mo>:</mo><mi>X</mi><mo>⊸</mo><mi>M</mi></math></span>, <span><math><mi>Φ</mi><mo>:</mo><mi>X</mi><mo>⊸</mo><mi>N</mi></math></span> are two admissible multi-valued maps and <em>N</em> is a connected closed manifold and a homology <em>n</em>-sphere.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109670"},"PeriodicalIF":0.5,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618328","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}
Pub Date : 2025-11-20DOI: 10.1016/j.topol.2025.109667
Richard N. Ball , Anthony W. Hager , Joanne Walters-Wayland
We show that classical continuous convergence has an elegant and powerful formulation in , the W-object of real functions on the locale L. Here W is the category of divisible archimedean ℓ-groups with designated weak order units, and cW is the category of W-objects equipped with a suitable convergence structure, together with the continuous W-homomorphisms. We exhibit an adjunction between the functor , which assigns to the locale L the cW-object equipped with its continuous convergence, and its left adjoint , which assigns to a given cW-object its frame of cW-kernels, i.e., its frame of convergence closed W-kernels. A significant attribute of the adjunction is that its unit is the Cauchy completion.
The suitable convergences, here termed admissibleW-convergences, include the classical convergences, are Hausdorff, and are all coarser than relative uniform convergence and finer than α-convergence. (Pointfree pointwise convergence, however, is not admissible.) We show that the coarse admissible convergences on a W-object G are in bijective correspondence with the dense nuclei on the frame of W-kernels of G. Finally, we show that the κ-closure nucleus on , i.e., the closure of the W-kernels of G under κ-joins, corresponds to the coarse admissible W-convergence on G whose Cauchy completion is the κ-repletion of G.
{"title":"From λ-hollow frames to λ-repletions in W: III. Continuous convergence in RL","authors":"Richard N. Ball , Anthony W. Hager , Joanne Walters-Wayland","doi":"10.1016/j.topol.2025.109667","DOIUrl":"10.1016/j.topol.2025.109667","url":null,"abstract":"<div><div>We show that classical continuous convergence has an elegant and powerful formulation in <span><math><mi>R</mi><mi>L</mi></math></span>, the <strong>W</strong>-object of real functions on the locale <em>L</em>. Here <strong>W</strong> is the category of divisible archimedean <em>ℓ</em>-groups with designated weak order units, and <strong>cW</strong> is the category of <strong>W</strong>-objects equipped with a suitable convergence structure, together with the continuous <strong>W</strong>-homomorphisms. We exhibit an adjunction between the functor <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>, which assigns to the locale <em>L</em> the <strong>cW</strong>-object <span><math><mi>R</mi><mi>L</mi></math></span> equipped with its continuous convergence, and its left adjoint <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>, which assigns to a given <strong>cW</strong>-object its frame of <strong>cW</strong>-kernels, i.e., its frame of convergence closed <strong>W</strong>-kernels. A significant attribute of the adjunction is that its unit is the Cauchy completion.</div><div>The suitable convergences, here termed <em>admissible</em> <strong>W</strong><em>-convergences</em>, include the classical convergences, are Hausdorff, and are all coarser than relative uniform convergence and finer than <em>α</em>-convergence. (Pointfree pointwise convergence, however, is not admissible.) We show that the coarse admissible convergences on a <strong>W</strong>-object <em>G</em> are in bijective correspondence with the dense nuclei on the frame <span><math><mi>K</mi><mi>G</mi></math></span> of <strong>W</strong>-kernels of <em>G</em>. Finally, we show that the <em>κ</em>-closure nucleus on <span><math><mi>K</mi><mi>G</mi></math></span>, i.e., the closure of the <strong>W</strong>-kernels of <em>G</em> under <em>κ</em>-joins, corresponds to the coarse admissible <strong>W</strong>-convergence on <em>G</em> whose Cauchy completion is the <em>κ</em>-repletion of <em>G</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109667"},"PeriodicalIF":0.5,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617865","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}
Pub Date : 2025-11-19DOI: 10.1016/j.topol.2025.109665
Phan Hoàng Chơn , Nguyễn Đặng Hồ Hải
N. Kuhn and C. Lloyd conjectured in [8] that the rank of the mod-2 Morava K-theory of the real Grassmannian manifolds could be written as a sum of products of binomial coefficients. The conjecture is easy for and was already checked by Kuhn and Lloyd for . The aim of this paper is to check this conjecture for the case .
n . Kuhn和C. Lloyd在[8]中推测，实格拉斯曼流形的mod2 Morava K-理论K(n)的秩可以写成二项式系数积的和。这个猜想在d=1时很容易，在d=2时已经被Kuhn和Lloyd验证过了。本文的目的是在d=3的情况下检验这个猜想。
{"title":"On the Morava K-theory of the real Grassmannian Gr3(Rm)","authors":"Phan Hoàng Chơn , Nguyễn Đặng Hồ Hải","doi":"10.1016/j.topol.2025.109665","DOIUrl":"10.1016/j.topol.2025.109665","url":null,"abstract":"<div><div>N. Kuhn and C. Lloyd conjectured in <span><span>[8]</span></span> that the rank of the mod-2 Morava K-theory <span><math><mi>K</mi><msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> of the real Grassmannian manifolds <span><math><mi>G</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></math></span> could be written as a sum of products of binomial coefficients. The conjecture is easy for <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span> and was already checked by Kuhn and Lloyd for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. The aim of this paper is to check this conjecture for the case <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109665"},"PeriodicalIF":0.5,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617866","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}
Pub Date : 2025-11-17DOI: 10.1016/j.topol.2025.109664
Ziteng Ma
We prove that a generic diffeomorphism of a closed surface is N-expansive for some positive integer N if and only if it is Anosov. Consequently, every closed surface different from the torus exhibits a residual subset of diffeomorphisms, none of which is N-expansive. This contrasts with a result of Artigue [2].
{"title":"N-expansiveness of generic surface diffeomorphisms","authors":"Ziteng Ma","doi":"10.1016/j.topol.2025.109664","DOIUrl":"10.1016/j.topol.2025.109664","url":null,"abstract":"<div><div>We prove that a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> generic diffeomorphism of a closed surface is <em>N</em>-expansive for some positive integer <em>N</em> if and only if it is Anosov. Consequently, every closed surface different from the torus exhibits a residual subset of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> diffeomorphisms, none of which is <em>N</em>-expansive. This contrasts with a result of Artigue <span><span>[2]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109664"},"PeriodicalIF":0.5,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571374","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}
Pub Date : 2025-11-14DOI: 10.1016/j.topol.2025.109661
Chunxu Ji , Kaiyun Wang
This paper focuses on the quasi covering dimension of topological spaces. We prove that a topological space and its Kolmogorov quotient space have the same quasi covering dimension. Subsequently, for a finite space , we show that its quasi covering dimension , where is the set of all maximal elements of X under the specialization order. Furthermore, we investigate the subspace theorems of quasi covering dimension of finite topological spaces and give an example presenting that the Cartesian product theorem does not hold in general. Finally, we discuss the relationships between the quasi covering dimension of topological spaces and that of frames.
{"title":"Quasi covering dimension of topological spaces","authors":"Chunxu Ji , Kaiyun Wang","doi":"10.1016/j.topol.2025.109661","DOIUrl":"10.1016/j.topol.2025.109661","url":null,"abstract":"<div><div>This paper focuses on the quasi covering dimension of topological spaces. We prove that a topological space and its Kolmogorov quotient space have the same quasi covering dimension. Subsequently, for a finite <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, we show that its quasi covering dimension <span><math><mi>d</mi><mi>i</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>w</mi><mi>i</mi><mi>d</mi><mi>t</mi><mi>h</mi><mo>(</mo><mo>↓</mo><mspace></mspace><mi>x</mi><mo>)</mo><mo>|</mo><mi>x</mi><mo>∈</mo><mrow><mi>Max</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mrow><mi>Max</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the set of all maximal elements of <em>X</em> under the specialization order. Furthermore, we investigate the subspace theorems of quasi covering dimension of finite topological spaces and give an example presenting that the Cartesian product theorem does not hold in general. Finally, we discuss the relationships between the quasi covering dimension of topological spaces and that of frames.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109661"},"PeriodicalIF":0.5,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571264","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}
Pub Date : 2025-11-13DOI: 10.1016/j.topol.2025.109662
Zhengbo Fang, Lei Mou
In this paper, we mainly prove that any countable product of ordinals is dually discrete. To get the result, we use the method which is called reverse induction by Patrakeev in [5].
{"title":"Any countable product of ordinals is dually discrete","authors":"Zhengbo Fang, Lei Mou","doi":"10.1016/j.topol.2025.109662","DOIUrl":"10.1016/j.topol.2025.109662","url":null,"abstract":"<div><div>In this paper, we mainly prove that any countable product of ordinals is dually discrete. To get the result, we use the method which is called reverse induction by Patrakeev in <span><span>[5]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109662"},"PeriodicalIF":0.5,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571373","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}
Pub Date : 2025-11-13DOI: 10.1016/j.topol.2025.109663
Abhishek
In 1971, R. D. Holmes and A. T.-M. Lau generalized the class of nonexpansive action by introducing the notion of asymptotically nonexpansive action of a semitopological semigroup S on a nonempty subset K of a Banach space U. They proved that a right reversible asymptotically nonexpansive action of a semitopological semigroup S on C has a common fixed point in C whenever C is a nonempty compact convex set having the property (B) in a Banach space U. In 2018, Aminpour et al. introduced the notion of strong asymptotically nonexpansive action of a semitopological semigroup S on a nonempty subset K of a Banach space U. Aminpour et al. gave an analogous result to R. D. Holmes and A. T.-M. Lau without the property (B). Without the assumption of property (B), we prove that a left reversible strong asymptotically nonexpansive action of a semitopological semigroup S on W has a common fixed point in W whenever W is a nonempty weakly compact convex set in a uniformly convex in every direction Banach space U. Moreover, it is proved that a strong asymptotically nonexpansive action of a semitopological semigroup S on W has a common fixed point in W whenever W is a nonempty weakly compact convex set having the normal structure in a Banach space U and S is compact and reversible. Under the assumption of the asymptotic center property, it is proved that a strong asymptotically nonexpansive action of a commutative semitopological semigroup S on E has a common fixed point in E whenever E is a nonempty weak⁎ compact convex set in a dual Banach space .
{"title":"Strong asymptotically nonexpansive action and common fixed points on weak and weak⁎ compact convex sets","authors":"Abhishek","doi":"10.1016/j.topol.2025.109663","DOIUrl":"10.1016/j.topol.2025.109663","url":null,"abstract":"<div><div>In 1971, R. D. Holmes and A. T.-M. Lau generalized the class of nonexpansive action by introducing the notion of asymptotically nonexpansive action of a semitopological semigroup S on a nonempty subset <em>K</em> of a Banach space <em>U</em>. They proved that a right reversible asymptotically nonexpansive action of a semitopological semigroup S on <em>C</em> has a common fixed point in <em>C</em> whenever <em>C</em> is a nonempty compact convex set having the property (B) in a Banach space <em>U</em>. In 2018, Aminpour et al. introduced the notion of strong asymptotically nonexpansive action of a semitopological semigroup S on a nonempty subset <em>K</em> of a Banach space <em>U</em>. Aminpour et al. gave an analogous result to R. D. Holmes and A. T.-M. Lau without the property (B). Without the assumption of property (B), we prove that a left reversible strong asymptotically nonexpansive action of a semitopological semigroup S on <em>W</em> has a common fixed point in <em>W</em> whenever <em>W</em> is a nonempty weakly compact convex set in a uniformly convex in every direction Banach space <em>U</em>. Moreover, it is proved that a strong asymptotically nonexpansive action of a semitopological semigroup S on <em>W</em> has a common fixed point in <em>W</em> whenever <em>W</em> is a nonempty weakly compact convex set having the normal structure in a Banach space <em>U</em> and S is compact and reversible. Under the assumption of the asymptotic center property, it is proved that a strong asymptotically nonexpansive action of a commutative semitopological semigroup S on <em>E</em> has a common fixed point in <em>E</em> whenever <em>E</em> is a nonempty weak<sup>⁎</sup> compact convex set in a dual Banach space <span><math><mi>U</mi><msup><mrow></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109663"},"PeriodicalIF":0.5,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571265","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}
Pub Date : 2025-11-11DOI: 10.1016/j.topol.2025.109658
Danica Kosanović
Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of in 4-manifolds. The majority of work goes into the case of framed immersed circles.
{"title":"On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold","authors":"Danica Kosanović","doi":"10.1016/j.topol.2025.109658","DOIUrl":"10.1016/j.topol.2025.109658","url":null,"abstract":"<div><div>Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> in 4-manifolds. The majority of work goes into the case of framed immersed circles.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109658"},"PeriodicalIF":0.5,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520225","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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