Pub Date : 2025-12-23DOI: 10.1016/j.topol.2025.109706
Felipe de J. Aguilar-Romero, David Herrera-Carrasco, Fernando Macías-Romero
Let X be a metric continuum, and let n be a positive integer. We denote by the hyperspace consisting of all nonempty closed subsets of X with at most n points. For , the n-fold symmetric product suspension of X is the quotient space . In this paper, we prove that if X is a meshed continuum, , and Y is a continuum such that is homeomorphic to , then X is homeomorphic to Y.
{"title":"Meshed continua have unique n-fold symmetric product suspension","authors":"Felipe de J. Aguilar-Romero, David Herrera-Carrasco, Fernando Macías-Romero","doi":"10.1016/j.topol.2025.109706","DOIUrl":"10.1016/j.topol.2025.109706","url":null,"abstract":"<div><div>Let <em>X</em> be a metric continuum, and let <em>n</em> be a positive integer. We denote by <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the hyperspace consisting of all nonempty closed subsets of <em>X</em> with at most <em>n</em> points. For <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span>, the <em>n-fold symmetric product suspension</em> of <em>X</em> is the quotient space <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In this paper, we prove that if <em>X</em> is a meshed continuum, <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, and <em>Y</em> is a continuum such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is homeomorphic to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo></math></span>, then <em>X</em> is homeomorphic to <em>Y</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109706"},"PeriodicalIF":0.5,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884994","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 : 2025-12-22DOI: 10.1016/j.topol.2025.109704
Donald M. Davis
It is known that, for all n, there exist compact differentiable orientable n-manifolds with dual Stiefel-Whitney class , and this is best possible, but the proof is nonconstructive. Here equals the number of 1's in the binary expansion of n if mod 4, and exceeds this by 1 otherwise. We find, for all mod 4, examples of real Bott manifolds with this property.
已知，对于所有n，存在紧可微可定向的n-流形，其对偶stiefell - whitney类w - n- α α - (n)≠0，这是最好的可能，但证明是非建设性的。如果n≡1 mod 4，则α - (n)等于n的二进制展开式中1的个数，否则超过1。我们找到了，对于所有n≥0 mod 4，具有这个性质的实博特流形的例子。
{"title":"Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading","authors":"Donald M. Davis","doi":"10.1016/j.topol.2025.109704","DOIUrl":"10.1016/j.topol.2025.109704","url":null,"abstract":"<div><div>It is known that, for all <em>n</em>, there exist compact differentiable orientable <em>n</em>-manifolds with dual Stiefel-Whitney class <span><math><msub><mrow><mover><mrow><mi>w</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>n</mi><mo>−</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msub><mo>≠</mo><mn>0</mn></math></span>, and this is best possible, but the proof is nonconstructive. Here <span><math><mover><mrow><mi>α</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>n</mi><mo>)</mo></math></span> equals the number of 1's in the binary expansion of <em>n</em> if <span><math><mi>n</mi><mo>≡</mo><mn>1</mn></math></span> mod 4, and exceeds this by 1 otherwise. We find, for all <span><math><mi>n</mi><mo>≢</mo><mn>0</mn></math></span> mod 4, examples of real Bott manifolds with this property.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109704"},"PeriodicalIF":0.5,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885041","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 : 2025-12-22DOI: 10.1016/j.topol.2025.109707
Alan Dow, Hayden Pecoraro
The property of selectively separable is well-studied and generalizations such as H-separable and wH-separable have also generated much interest. Bardyla, Maesano, and Zdomskyy proved from Martin's Axiom that there are countable regular wH-separable spaces that are not H-separable. We prove there is a ZFC example. Their example was also Fréchet-Urysohn, and we produce two additional examples from weaker assumptions.
{"title":"New examples in the study of selectively separable spaces","authors":"Alan Dow, Hayden Pecoraro","doi":"10.1016/j.topol.2025.109707","DOIUrl":"10.1016/j.topol.2025.109707","url":null,"abstract":"<div><div>The property of selectively separable is well-studied and generalizations such as H-separable and wH-separable have also generated much interest. Bardyla, Maesano, and Zdomskyy proved from Martin's Axiom that there are countable regular wH-separable spaces that are not H-separable. We prove there is a ZFC example. Their example was also Fréchet-Urysohn, and we produce two additional examples from weaker assumptions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109707"},"PeriodicalIF":0.5,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884991","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 : 2025-12-18DOI: 10.1016/j.topol.2025.109694
Prerak Deep, Dheeraj Kulkarni
We introduce the notion of contact round surgery of index 1 on Legendrian knots in a general contact 3-manifold. It generalizes the notion of contact round surgery of index 1 on Legendrian knots introduced by Adachi. In , we introduce the notion of contact round surgery of index 2 on a Legendrian knot and realize Adachi's contact round 2-surgery on a convex torus as a contact round surgery of index 2 on a Legendrian knot in . We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in . With this set up, we show that every closed connected contact 3-manifold can be obtained by performing a sequence of contact round surgeries on some Legendrian link in , thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to the result of Ding-Geiges for contact Dehn surgeries. We also discuss a bridge between certain pairs of contact round surgery diagrams of indices 1 and 2, and contact -surgery diagrams. We use this bridge to establish the result mentioned above. In the end, we derive a corollary that gives sufficient conditions on contact round surgeries to produce symplectically fillable manifolds.
{"title":"On contact round surgeries on (S3,ξst) and their diagrams","authors":"Prerak Deep, Dheeraj Kulkarni","doi":"10.1016/j.topol.2025.109694","DOIUrl":"10.1016/j.topol.2025.109694","url":null,"abstract":"<div><div>We introduce the notion of contact round surgery of index 1 on Legendrian knots in a general contact 3-manifold. It generalizes the notion of contact round surgery of index 1 on Legendrian knots introduced by Adachi. In <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>)</mo></math></span>, we introduce the notion of contact round surgery of index 2 on a Legendrian knot and realize Adachi's contact round 2-surgery on a convex torus as a contact round surgery of index 2 on a Legendrian knot in <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>)</mo></math></span>. We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>)</mo></math></span>. With this set up, we show that every closed connected contact 3-manifold can be obtained by performing a sequence of contact round surgeries on some Legendrian link in <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>)</mo></math></span>, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to the result of Ding-Geiges for contact Dehn surgeries. We also discuss a bridge between certain pairs of contact round surgery diagrams of indices 1 and 2, and contact <span><math><mo>(</mo><mo>±</mo><mn>1</mn><mo>)</mo></math></span>-surgery diagrams. We use this bridge to establish the result mentioned above. In the end, we derive a corollary that gives sufficient conditions on contact round surgeries to produce symplectically fillable manifolds.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109694"},"PeriodicalIF":0.5,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841749","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 : 2025-12-16DOI: 10.1016/j.topol.2025.109691
Tsukasa Isoshima
We show that two kinds of trisection diagrams for the doubles of the Mazur type 4-manifolds introduced by Akbulut and Kirby are standard. One is constructed by doubling a certain relative trisection diagram of the Mazur type 4-manifold. The other is constructed using an algorithm for taking Kirby diagrams to trisection diagrams.
{"title":"Trisections of the doubles of some Mazur type 4-manifolds","authors":"Tsukasa Isoshima","doi":"10.1016/j.topol.2025.109691","DOIUrl":"10.1016/j.topol.2025.109691","url":null,"abstract":"<div><div>We show that two kinds of trisection diagrams for the doubles of the Mazur type 4-manifolds introduced by Akbulut and Kirby are standard. One is constructed by doubling a certain relative trisection diagram of the Mazur type 4-manifold. The other is constructed using an algorithm for taking Kirby diagrams to trisection diagrams.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109691"},"PeriodicalIF":0.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841750","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 : 2025-12-16DOI: 10.1016/j.topol.2025.109693
Jelena Katić, Darko Milinković
It is known that Morse-Smale diffeomorphisms have the shadowing property; however, the question of whether also has the shadowing property when f is Morse-Smale remains open and has been resolved only in a few specific cases [3]. We prove that if is a time-one-map of Morse gradient flow, the induced map on the hyperspace of subcontinua does not have the shadowing property.
{"title":"Shadowing property on hyperspace of continua induced by Morse gradient system","authors":"Jelena Katić, Darko Milinković","doi":"10.1016/j.topol.2025.109693","DOIUrl":"10.1016/j.topol.2025.109693","url":null,"abstract":"<div><div>It is known that Morse-Smale diffeomorphisms have the shadowing property; however, the question of whether <span><math><mi>C</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> also has the shadowing property when <em>f</em> is Morse-Smale remains open and has been resolved only in a few specific cases <span><span>[3]</span></span>. We prove that if <span><math><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></math></span> is a time-one-map of Morse gradient flow, the induced map <span><math><mi>C</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><mi>C</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>→</mo><mi>C</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> on the hyperspace of subcontinua does not have the shadowing property.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109693"},"PeriodicalIF":0.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841751","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 : 2025-12-15DOI: 10.1016/j.topol.2025.109692
Ravi Tomar
In this paper, we introduce the notion of relatively acylindrical action for a graph of relatively hyperbolic groups. We then prove a combination theorem for relatively acylindrical graphs of relatively hyperbolic groups, which generalizes Dahmani's combination theorem for acylindrical graphs of relatively hyperbolic groups. Finally, we deduce some applications of this result.
{"title":"A combination theorem for relatively acylindrical graphs of relatively hyperbolic groups","authors":"Ravi Tomar","doi":"10.1016/j.topol.2025.109692","DOIUrl":"10.1016/j.topol.2025.109692","url":null,"abstract":"<div><div>In this paper, we introduce the notion of relatively acylindrical action for a graph of relatively hyperbolic groups. We then prove a combination theorem for relatively acylindrical graphs of relatively hyperbolic groups, which generalizes Dahmani's combination theorem for acylindrical graphs of relatively hyperbolic groups. Finally, we deduce some applications of this result.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109692"},"PeriodicalIF":0.5,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792228","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 : 2025-12-10DOI: 10.1016/j.topol.2025.109688
Changchun Xia
The main purpose of this paper is to investigate the extensions of -convex spaces and further to study the right Kan convex spaces from the viewpoints of classical convexity theory and lattice-theoretic approach. Firstly, we show that the strict (strictly dense) extensions of an -convex space are completely determined by the convex subspaces of ΦX () containing all the principal Scott closed subsets of , up to convex-homeomorphism, where () ΦX is the set of all the (proper) Scott closed subsets of ; Secondly, we introduce the notion of right Kan convex spaces and present several necessary and sufficient conditions for -convex spaces to be right Kan; Moreover, we show that the set of all the dense Scott closed subsets of an -convex space X as a convex subspace of ΦX is essential in the category of -convex spaces, but not an injective hull of X in general; Finally, from the lattice-theoretic approach, by introducing the notion of convex elements of a continuous lattice L, we show that L equipped with the convex structure generated by the family as a subbase is a right Kan convex space iff every element of L is convex and build a relationship between the convex elements and Scott closed subsets of L. In particular, we show that a convex subset of X is a convergence set iff it is a convex element of .
{"title":"Characterization of right Kan convex spaces via domain theory","authors":"Changchun Xia","doi":"10.1016/j.topol.2025.109688","DOIUrl":"10.1016/j.topol.2025.109688","url":null,"abstract":"<div><div>The main purpose of this paper is to investigate the extensions of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-convex spaces and further to study the right Kan convex spaces from the viewpoints of classical convexity theory and lattice-theoretic approach. Firstly, we show that the strict (strictly dense) extensions of an <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-convex space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> are completely determined by the convex subspaces of Φ<em>X</em> (<span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mi>o</mi></mrow></msup><mi>X</mi></math></span>) containing all the principal Scott closed subsets of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>, up to convex-homeomorphism, where (<span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mi>o</mi></mrow></msup><mi>X</mi></math></span>) Φ<em>X</em> is the set of all the (proper) Scott closed subsets of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>; Secondly, we introduce the notion of right Kan convex spaces and present several necessary and sufficient conditions for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-convex spaces to be right Kan; Moreover, we show that the set of all the dense Scott closed subsets of an <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-convex space <em>X</em> as a convex subspace of Φ<em>X</em> is essential in the category of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-convex spaces, but not an injective hull of <em>X</em> in general; Finally, from the lattice-theoretic approach, by introducing the notion of convex elements of a continuous lattice <em>L</em>, we show that <em>L</em> equipped with the convex structure generated by the family <span><math><mo>{</mo><mo>⇓</mo><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>L</mi><mo>}</mo></math></span> as a subbase is a right Kan convex space iff every element of <em>L</em> is convex and build a relationship between the convex elements and Scott closed subsets of <em>L</em>. In particular, we show that a convex subset of <em>X</em> is a convergence set iff it is a convex element of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109688"},"PeriodicalIF":0.5,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790740","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 : 2025-12-10DOI: 10.1016/j.topol.2025.109690
Domagoj Jelić , Piotr Oprocha
The paper studies the structure of ω-limit sets of map induced on the hyperspace of all connected compact sets, by dynamical system acting on a topological graph G. In the case of the base space being a topological tree we additionally show that is always almost equicontinuous and characterize its Birkhoff center.
{"title":"On limit sets and equicontinuity in the hyperspace of continua in dimension one","authors":"Domagoj Jelić , Piotr Oprocha","doi":"10.1016/j.topol.2025.109690","DOIUrl":"10.1016/j.topol.2025.109690","url":null,"abstract":"<div><div>The paper studies the structure of <em>ω</em>-limit sets of map <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> induced on the hyperspace <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of all connected compact sets, by dynamical system <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> acting on a topological graph <em>G</em>. In the case of the base space being a topological tree we additionally show that <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> is always almost equicontinuous and characterize its Birkhoff center.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109690"},"PeriodicalIF":0.5,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760745","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 : 2025-12-09DOI: 10.1016/j.topol.2025.109687
Mykola Lysynskyi, Sergiy Maksymenko
In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold called the line with two origins which is obtained by gluing two copies of the real line via the identity homeomorphism of .
Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold (called letter “Y”) obtained by gluing two copies of via the identity map of positive reals. It turns out that, in contrast to the real line, for every , both manifolds and admit uncountably many pair-wise non-diffeomorphic -structures.
We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.
{"title":"Differentiable structures on a union of two open sets","authors":"Mykola Lysynskyi, Sergiy Maksymenko","doi":"10.1016/j.topol.2025.109687","DOIUrl":"10.1016/j.topol.2025.109687","url":null,"abstract":"<div><div>In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold <span><math><mi>L</mi></math></span> called the <em>line with two origins</em> which is obtained by gluing two copies of the real line <span><math><mi>R</mi></math></span> via the identity homeomorphism of <span><math><mi>R</mi><mo>∖</mo><mn>0</mn></math></span>.</div><div>Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold <span><math><mi>Y</mi></math></span> (called <em>letter</em> “<em>Y</em>”) obtained by gluing two copies of <span><math><mi>R</mi></math></span> via the identity map of positive reals. It turns out that, in contrast to the real line, for every <span><math><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>∞</mo></math></span>, both manifolds <span><math><mi>L</mi></math></span> and <span><math><mi>Y</mi></math></span> admit uncountably many pair-wise non-diffeomorphic <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>-structures.</div><div>We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109687"},"PeriodicalIF":0.5,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790739","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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