Topology and its Applications最新文献

{"title":"Ultrafilters and the Katětov order","authors":"Krzysztof Kowitz,&nbsp;Adam Kwela","doi":"10.1016/j.topol.2024.109191","DOIUrl":"10.1016/j.topol.2024.109191","url":null,"abstract":"<div><div>Let <span><math><mi>I</mi></math></span> be an ideal on <em>ω</em>. Following Baumgartner (1995) <span><span>[2]</span></span>, we say that an ultrafilter <span><math><mi>U</mi></math></span> on <em>ω</em> is an <span><math><mi>I</mi></math></span>-ultrafilter if for every function <span><math><mi>f</mi><mo>:</mo><mi>ω</mi><mo>→</mo><mi>ω</mi></math></span> there is <span><math><mi>A</mi><mo>∈</mo><mi>U</mi></math></span> with <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo><mo>∈</mo><mi>I</mi></math></span>. In particular, P-points are exactly <span><math><mrow><mi>Fin</mi></mrow><mo>×</mo><mrow><mi>Fin</mi></mrow></math></span>-ultrafilters.</div><div>If there is an <span><math><mi>I</mi></math></span>-ultrafilter which is not a <span><math><mi>J</mi></math></span>-ultrafilter, then <span><math><mi>I</mi></math></span> is not below <span><math><mi>J</mi></math></span> in the Katětov order <span><math><msub><mrow><mo>⩽</mo></mrow><mrow><mi>K</mi></mrow></msub></math></span> (i.e., for every function <span><math><mi>f</mi><mo>:</mo><mi>ω</mi><mo>→</mo><mi>ω</mi></math></span> there is <span><math><mi>A</mi><mo>∈</mo><mi>I</mi></math></span> with <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>[</mo><mi>A</mi><mo>]</mo><mo>∉</mo><mi>J</mi></math></span>), however the reversed implication is not true (even consistently).</div><div>Recently it was shown that for all Borel ideals <span><math><mi>I</mi></math></span> we have: <span><math><mi>I</mi><msub><mrow><mo>≰</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mi>Fin</mi></mrow><mo>×</mo><mrow><mi>Fin</mi></mrow></math></span> if and only if in some forcing extension one can find an <span><math><mi>I</mi></math></span>-ultrafilter which is not a P-point (Filipów et al. (2022) <span><span>[6]</span></span>).</div><div>We show that under some combinatorial assumptions imposed on the ideal <span><math><mi>J</mi></math></span>, the classes of <span><math><mi>J</mi></math></span>-ultrafilters and <span><math><mrow><mi>Fin</mi></mrow><mo>×</mo><mi>J</mi></math></span>-ultrafilters coincide. This allows us to find some sufficient conditions on ideals to obtain the equivalence: <span><math><mi>I</mi><msub><mrow><mo>≰</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mi>Fin</mi></mrow><mo>×</mo><mi>J</mi></math></span> if and only if in some forcing extension one can find an <span><math><mi>I</mi></math></span>-ultrafilter which is not a <span><math><mi>J</mi></math></span>-ultrafilter. We provide several examples of ideals, for which the above equivalence is true, including the ideal of nowhere dense subsets of <span><math><mi>Q</mi></math></span> and the ideal of sets of asymptotic density zero.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109191"},"PeriodicalIF":0.6,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146940","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":"End-essential spanning surfaces for links in thickened surfaces","authors":"Thomas Kindred","doi":"10.1016/j.topol.2024.109187","DOIUrl":"10.1016/j.topol.2024.109187","url":null,"abstract":"<div><div>Let <em>D</em> be a cellular alternating link diagram on a closed orientable surface Σ. We prove that if <em>D</em> has no removable nugatory crossings then each checkerboard surface from <em>D</em> is <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-essential and contains no essential closed curve that is ∂-parallel in <span><math><mi>Σ</mi><mo>×</mo><mi>I</mi></math></span>. Our chief motivation comes from technical aspects of a companion paper, where we prove that Tait's flyping conjecture holds for alternating virtual links. We also describe possible applications via Turaev surfaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109187"},"PeriodicalIF":0.6,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147417","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":"Satellite operations and the loop expansion of the Kontsevich invariant of knots","authors":"Kouki Yamaguchi","doi":"10.1016/j.topol.2024.109189","DOIUrl":"10.1016/j.topol.2024.109189","url":null,"abstract":"<div><div>The Kontsevich invariant of knots has a special expansion, which is called the loop expansion. In this paper, we present the behavior of some loop parts, after some types of satellite operation; this types of satellite operation do not change the 1-loop part, the Alexander polynomial.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109189"},"PeriodicalIF":0.6,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147421","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":"Extending periodic maps from the genus 2 surface to the 3-torus","authors":"Weibiao Wang ,&nbsp;Yimu Zhang","doi":"10.1016/j.topol.2024.109186","DOIUrl":"10.1016/j.topol.2024.109186","url":null,"abstract":"<div><div>There are 22 nontrivial periodic maps on the orientable closed surface of genus 2, up to conjugacy. We determine whether they can extend to periodic maps on the 3-torus, orientation-preservingly or orientation-reversingly.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109186"},"PeriodicalIF":0.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146941","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":"Dichotomy theorem for topological semigroup actions","authors":"Zhumin Ding ,&nbsp;Yu Huang ,&nbsp;Tao Wang","doi":"10.1016/j.topol.2024.109188","DOIUrl":"10.1016/j.topol.2024.109188","url":null,"abstract":"<div><div>We introduce the notions of equi-stable points and equi-asymptotically stable points for topological semigroup actions with a regular system and explore the totally different behaviors of control sets based on these two kinds of points, which are the analogies to the well-known dichotomy theorem for topological transitive dynamical systems. Besides, two illustrative examples are given to show that the equi-asymptotically stable points are different from equi-stable points.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109188"},"PeriodicalIF":0.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146802","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":"Constrained motion spaces of robotic arms","authors":"Jack Pierce","doi":"10.1016/j.topol.2024.109184","DOIUrl":"10.1016/j.topol.2024.109184","url":null,"abstract":"<div><div>In this paper, we develop the theory of constrained motion spaces of robotic arms. We compute their homology groups in two cases: when the constraint is a horizontal line and when it is a smooth curve whose motion space is a smooth manifold. We show the computation of homology amounts to counting the collinear configurations, reducing a topological problem to a combinatorial problem. Our results rely on Morse theory, along with Walker's and Farber's work on polygonal linkages.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109184"},"PeriodicalIF":0.6,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147418","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 forms of shadowing and stability for set-valued maps","authors":"Abdul Gaffar Khan ,&nbsp;Ramesh Kumar ,&nbsp;Tarun Das","doi":"10.1016/j.topol.2024.109182","DOIUrl":"10.1016/j.topol.2024.109182","url":null,"abstract":"<div><div>In this paper, we introduce the notion of iterated product space and define positive <em>i</em>-shadowing property and cw<em>i</em>-topological stability for upper-semicontinuous closed-valued maps. We prove that every map with the positive pseudo-orbit tracing property has the positive <em>i</em>-shadowing property but there exists a map on the unit circle which has the positive <em>i</em>-shadowing property but does not have the positive pseudo-orbit tracing property. We also provide an example of a cw<em>i</em>-topologically stable set-valued map which is not cw-topologically stable. We provide sufficient conditions under which cw<em>i</em>-topological stability is equivalent to positive <em>i</em>-shadowing property. We further prove that total transitivity and topologically weakly mixing are equivalent notions for continuous onto closed-valued maps having positive <em>i</em>-shadowing property.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"361 ","pages":"Article 109182"},"PeriodicalIF":0.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147416","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":"Cartesian product of combinatorially rich sets- algebraic, elementary and dynamical approaches","authors":"Pintu Debnath","doi":"10.1016/j.topol.2024.109148","DOIUrl":"10.1016/j.topol.2024.109148","url":null,"abstract":"<div><div>Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. In [Fund. Math 199 (2008)], D. De, N. Hindman, and D. Strauss introduced the notion of a <em>C</em>-set, satisfying the strong central sets theorem. In [Topology Proc. 35 (2010)], using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two <em>C</em>-sets is a <em>C</em>-set. S. Goswami has proved the same result using the elementary characterization of <em>C</em>-sets. In this article, we will prove that the product of two <em>C</em>-sets is a <em>C</em>-set, using the dynamical characterization of <em>C</em>-sets. Recently, S. Goswami has proved that the Cartesian product of two <em>CR</em>-sets is a <em>CR</em>-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential <em>CR</em>-sets is an essential <em>CR</em>-set.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109148"},"PeriodicalIF":0.6,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Rudin-Kiesler pre-order and the Pixley-Roy spaces over ultrafilters","authors":"Masami Sakai","doi":"10.1016/j.topol.2024.109136","DOIUrl":"10.1016/j.topol.2024.109136","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-space &lt;em&gt;X&lt;/em&gt;, we denote by &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; the Pixley-Roy space over &lt;em&gt;X&lt;/em&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be the subspace of the Stone-Čech compactification &lt;em&gt;βω&lt;/em&gt; of the discrete space &lt;em&gt;ω&lt;/em&gt;. Motivated by Gul'ko's theorem (&lt;span&gt;&lt;span&gt;Theorem 1.1&lt;/span&gt;&lt;/span&gt;), we show: (1) &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic if and only if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic (i.e., &lt;em&gt;p&lt;/em&gt; and &lt;em&gt;q&lt;/em&gt; are type-equivalent), (2) if &lt;em&gt;q&lt;/em&gt; is selective and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; can be embedded into &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic, (3) if &lt;em&gt;p&lt;/em&gt; is selective, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; contains copies of some &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; which are pairwise non-homeomorphic, and (4) &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are pairwise non-homeomorphic, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the quotient space obtained by identifying the limit points of the topological sum &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mro","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109136"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Dold-Whitney's parallelizability of 4-manifolds","authors":"Valentina Bais","doi":"10.1016/j.topol.2024.109144","DOIUrl":"10.1016/j.topol.2024.109144","url":null,"abstract":"<div><div>We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109144"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142719800","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}