Pub Date : 2025-10-17DOI: 10.1016/j.topol.2025.109638
Mitsuyoshi Adachi
In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy K3 surface , if the tangent bundle along the fibers admits a spin structure, then also admits a spin structure, where is the vector bundle consisting of self-dual harmonic 2-forms. In this paper, we show that admits a canonical spin structure. The proof is carried out by canonically constructing a lifting -gerbe for the spin structure on using the families Seiberg–Witten equations, starting from a lifting -gerbe for the spin structure on .
{"title":"A gerbe-like construction in gauge theory","authors":"Mitsuyoshi Adachi","doi":"10.1016/j.topol.2025.109638","DOIUrl":"10.1016/j.topol.2025.109638","url":null,"abstract":"<div><div>In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy <em>K</em>3 surface <span><math><mi>X</mi><mo>→</mo><mi>X</mi><mover><mrow><mo>→</mo></mrow><mrow><mi>π</mi></mrow></mover><mi>B</mi></math></span>, if the tangent bundle along the fibers <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>X</mi></math></span> admits a spin structure, then <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> also admits a spin structure, where <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the vector bundle consisting of self-dual harmonic 2-forms. In this paper, we show that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>X</mi><mo>⊕</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup><msup><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> admits a canonical spin structure. The proof is carried out by canonically constructing a lifting <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-gerbe for the spin structure on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> using the families Seiberg–Witten equations, starting from a lifting <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-gerbe for the spin structure on <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>X</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109638"},"PeriodicalIF":0.5,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364657","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 : 2025-10-13DOI: 10.1016/j.topol.2025.109636
Khulod Almontashery , Paul J. Szeptycki
We introduce a strengthening of the class of the proximal and semi-proximal spaces by restricting the proximal game to totally bounded uniformities. In addition, we examine the connections between the proximal game and two well-known games, one set-theoretic the other topological: the Galvin game and the Gruenhage game.
{"title":"The proximal game and its connections to other games","authors":"Khulod Almontashery , Paul J. Szeptycki","doi":"10.1016/j.topol.2025.109636","DOIUrl":"10.1016/j.topol.2025.109636","url":null,"abstract":"<div><div>We introduce a strengthening of the class of the proximal and semi-proximal spaces by restricting the proximal game to totally bounded uniformities. In addition, we examine the connections between the proximal game and two well-known games, one set-theoretic the other topological: the Galvin game and the Gruenhage game.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109636"},"PeriodicalIF":0.5,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321827","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 : 2025-10-10DOI: 10.1016/j.topol.2025.109634
Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee
In this paper, we present a framework for discrete motion planning tailored for robots that operate in a discrete manner. Furthermore, we extend the concept of r-discrete homotopy as discrete -homotopy. Utilizing this framework, we investigate the notion of discrete topological complexity, which is defined as the least number of motion planning algorithms necessary for discrete movement. We establish several properties related to discrete topological complexity; for example, we demonstrate that discrete motion planning within a metric space X is feasible if and only if X is a discrete contractible space. Additionally, we show that the discrete topological complexity is solely determined by the strictly discrete homotopy type of the spaces involved.
{"title":"A discrete topological complexity of discrete motion planning","authors":"Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee","doi":"10.1016/j.topol.2025.109634","DOIUrl":"10.1016/j.topol.2025.109634","url":null,"abstract":"<div><div>In this paper, we present a framework for discrete motion planning tailored for robots that operate in a discrete manner. Furthermore, we extend the concept of <em>r</em>-discrete homotopy as discrete <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-homotopy. Utilizing this framework, we investigate the notion of discrete topological complexity, which is defined as the least number of motion planning algorithms necessary for discrete movement. We establish several properties related to discrete topological complexity; for example, we demonstrate that discrete motion planning within a metric space <em>X</em> is feasible if and only if <em>X</em> is a discrete contractible space. Additionally, we show that the discrete topological complexity is solely determined by the strictly discrete homotopy type of the spaces involved.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109634"},"PeriodicalIF":0.5,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269844","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 : 2025-10-10DOI: 10.1016/j.topol.2025.109635
Ajit Kumar Gupta , Saikat Mukherjee
We define two properties for subsets of a metric space. One of them is a generalization of chainability, finite chainability, and Menger convexity for metric spaces; and the other extends the notion of compactness for subsets of a metric space. We establish several fundamental results concerning these two properties. Further, in the context of these properties, we study the Hausdorff metric and derive the relations among Hausdorff, Vietoris, and locally finite hypertopologies on the collection of nonempty closed subsets of a metric space.
{"title":"Generalizations of chainability and compactness, and the hypertopologies","authors":"Ajit Kumar Gupta , Saikat Mukherjee","doi":"10.1016/j.topol.2025.109635","DOIUrl":"10.1016/j.topol.2025.109635","url":null,"abstract":"<div><div>We define two properties for subsets of a metric space. One of them is a generalization of chainability, finite chainability, and Menger convexity for metric spaces; and the other extends the notion of compactness for subsets of a metric space. We establish several fundamental results concerning these two properties. Further, in the context of these properties, we study the Hausdorff metric and derive the relations among Hausdorff, Vietoris, and locally finite hypertopologies on the collection of nonempty closed subsets of a metric space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109635"},"PeriodicalIF":0.5,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321824","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 : 2025-10-10DOI: 10.1016/j.topol.2025.109633
Yongqiang Liu , Wentao Xie
In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik [3, Theorem 1]. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams [36, Main Theorem 1] for complexified line arrangement to any complex line arrangement.
本文研究了复线排列补的有限循环覆盖的第一同调群。在与Cohen-Dimca-Orlik[3，定理1]相似的条件下，证明了第一个积分同调群是无扭转的。特别地，这包括Milnor纤维的情况，它将Williams [36， Main Theorem 1]先前得到的关于复线排列的结果推广到任何复线排列。
{"title":"The homology groups of finite cyclic coverings of line arrangement complements","authors":"Yongqiang Liu , Wentao Xie","doi":"10.1016/j.topol.2025.109633","DOIUrl":"10.1016/j.topol.2025.109633","url":null,"abstract":"<div><div>In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik <span><span>[3, Theorem 1]</span></span>. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams <span><span>[36, Main Theorem 1]</span></span> for complexified line arrangement to any complex line arrangement.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109633"},"PeriodicalIF":0.5,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321825","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 : 2025-10-06DOI: 10.1016/j.topol.2025.109631
Nathan Carlson
We use the notion of an o-free sequence to give new bounds for the cardinality of Hausdorff spaces and regular spaces. There are several implications for compacta. One is that if X is a compactum then , where is the o-tightness introduced by Tkachenko. Another is that if X is a compactum. This is shown to be a strict improvement of Arhangel'skiĭ's bound . Finally, we show if X is a homogeneous compactum. We note for such spaces, where is de la Vega's bound for the cardinality of homogeneous compacta.
我们利用无0序列的概念给出了Hausdorff空间和正则空间的基数的新边界。compact有几个含义。一是如果X是紧致，则w(X)≤hL(X)ot(X)，其中ot(X)为Tkachenko引入的o紧性。另一个是|X|≤hL(X)ot(X)wψc(X)，如果X是紧致的。这被证明是Arhangel'ski 's界2ψ(X)的严格改进。最后，我们证明了如果X是齐次紧致，|X|≤hL(X)ot(X)πχ(X)。我们注意到hL(X)ot(X)πχ(X)≤2t(X)，其中2t(X)是齐次紧的基数的de la Vega界。
{"title":"On o-free sequences and compacta","authors":"Nathan Carlson","doi":"10.1016/j.topol.2025.109631","DOIUrl":"10.1016/j.topol.2025.109631","url":null,"abstract":"<div><div>We use the notion of an <em>o</em>-free sequence to give new bounds for the cardinality of Hausdorff spaces and regular spaces. There are several implications for compacta. One is that if <em>X</em> is a compactum then <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>, where <span><math><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the <em>o</em>-tightness introduced by Tkachenko. Another is that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>w</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a compactum. This is shown to be a strict improvement of Arhangel'skiĭ's bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>. Finally, we show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a homogeneous compactum. We note <span><math><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for such spaces, where <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> is de la Vega's bound for the cardinality of homogeneous compacta.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109631"},"PeriodicalIF":0.5,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269842","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":"Weak approximation by points in function spaces and in the power of Arens' space","authors":"Kenichi Tamano , Stevo Todorčević","doi":"10.1016/j.topol.2025.109629","DOIUrl":"10.1016/j.topol.2025.109629","url":null,"abstract":"<div><div>We study the weak approximation by points (WAP) in function spaces <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and in the power <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msubsup></math></span> of Arens' space <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. The following two results are shown:</div><div>(1) The space <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msubsup></math></span>, which can be embedded in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>, is WAP, answering a question of G. Gruenhage, B. Tsaban, and L. Zdomskyy.</div><div>(2) <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span> is not WAP.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109629"},"PeriodicalIF":0.5,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321826","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 : 2025-10-03DOI: 10.1016/j.topol.2025.109630
Antoni Machowski
We examine subgroups of locally compact groups that are continuous homomorphic images of connected Lie groups and we give a criterion for being such an image. We also provide a new characterisation of Lie groups and a characterisation of groups that are images of connected locally compact groups.
{"title":"Virtual Lie subgroups of locally compact groups","authors":"Antoni Machowski","doi":"10.1016/j.topol.2025.109630","DOIUrl":"10.1016/j.topol.2025.109630","url":null,"abstract":"<div><div>We examine subgroups of locally compact groups that are continuous homomorphic images of connected Lie groups and we give a criterion for being such an image. We also provide a new characterisation of Lie groups and a characterisation of groups that are images of connected locally compact groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109630"},"PeriodicalIF":0.5,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145322260","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 : 2025-10-02DOI: 10.1016/j.topol.2025.109601
Mengqiao Huang , Xiaodong Jia , Qingguo Li
For a weak partial metric space , there is a canonical metric on X, defined as for all . We prove that the partial metric topology and the Scott topology on coincide if and only if the metric topology on and the Lawson topology on agree, provided that the weak partial metric space is a domain in its specialization order and its associated metric space is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.
{"title":"Topologies and fixpoints on weak partial metric spaces","authors":"Mengqiao Huang , Xiaodong Jia , Qingguo Li","doi":"10.1016/j.topol.2025.109601","DOIUrl":"10.1016/j.topol.2025.109601","url":null,"abstract":"<div><div>For a weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, there is a canonical metric <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> on <em>X</em>, defined as <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. We prove that the partial metric topology and the Scott topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> coincide if and only if the metric topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> and the Lawson topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> agree, provided that the weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a domain in its specialization order and its associated metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109601"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269836","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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