Journal of Topology最新文献

{"title":"Coarse cubical rigidity","authors":"Elia Fioravanti,&nbsp;Ivan Levcovitz,&nbsp;Michah Sageev","doi":"10.1112/topo.12353","DOIUrl":"https://doi.org/10.1112/topo.12353","url":null,"abstract":"<p>We show that for many right-angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non-hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$geqslant 3$</annotation>\u0000 </semantics></math>, we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12353","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Degenerations of \u0000 \u0000 k\u0000 $k$\u0000 -positive surface group representations","authors":"Jonas Beyrer,&nbsp;Beatrice Pozzetti","doi":"10.1112/topo.12352","DOIUrl":"https://doi.org/10.1112/topo.12352","url":null,"abstract":"<p>We introduce <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-<i>positive representations</i>, a large class of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace 1,ldots ,krbrace$</annotation>\u0000 </semantics></math>-Anosov surface group representations into <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>PGL</mi>\u0000 <mo>(</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathsf {PGL}(E)$</annotation>\u0000 </semantics></math> that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k-3)$</annotation>\u0000 </semantics></math>-positive and irreducible limits are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k-1)$</annotation>\u0000 </semantics></math>-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homeomorphism groups of 2-manifolds with the virtual Rokhlin property","authors":"Justin Lanier,&nbsp;Nicholas G. Vlamis","doi":"10.1112/topo.12354","DOIUrl":"https://doi.org/10.1112/topo.12354","url":null,"abstract":"<p>We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Applications of higher-dimensional Heegaard Floer homology to contact topology","authors":"Vincent Colin,&nbsp;Ko Honda,&nbsp;Yin Tian","doi":"10.1112/topo.12349","DOIUrl":"https://doi.org/10.1112/topo.12349","url":null,"abstract":"<p>The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Characteristic cohomology II: Matrix singularities","authors":"James Damon","doi":"10.1112/topo.12330","DOIUrl":"https://doi.org/10.1112/topo.12330","url":null,"abstract":"&lt;p&gt;For a germ of a variety &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}, 0 subset mathbb {C}^N, 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a singularity &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathcal {V}_0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of “type &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {V}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;”  is given by a germ &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$f_0: mathbb {C}^n, 0 rightarrow mathbb {C}^N, 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, which is transverse to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∖&lt;/mo&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}setminus lbrace 0rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in an appropriate sense, such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}_0 = f_0^{-1}(mathcal {V})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local connectedness of boundaries for relatively hyperbolic groups","authors":"Ashani Dasgupta,&nbsp;G. Christopher Hruska","doi":"10.1112/topo.12347","DOIUrl":"https://doi.org/10.1112/topo.12347","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Γ</mi>\u0000 <mo>,</mo>\u0000 <mi>P</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Gamma,mathbb {P})$</annotation>\u0000 </semantics></math> be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Γ</mi>\u0000 <mo>,</mo>\u0000 <mi>P</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Gamma,mathbb {P})$</annotation>\u0000 </semantics></math> is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of <span></span><math>\u0000 <semantics>\u0000 <mi>Γ</mi>\u0000 <annotation>$Gamma$</annotation>\u0000 </semantics></math> and no restriction on the peripheral subgroups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>∈</mo>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation>$P in mathbb {P}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Involutions, links, and Floer cohomologies","authors":"Hokuto Konno,&nbsp;Jin Miyazawa,&nbsp;Masaki Taniguchi","doi":"10.1112/topo.12340","DOIUrl":"https://doi.org/10.1112/topo.12340","url":null,"abstract":"<p>We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>spin</mi>\u0000 <mi>c</mi>\u0000 </msup>\u0000 <annotation>${rm spin}^c$</annotation>\u0000 </semantics></math> 4-manifold with boundary and with an involution that reverses the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>spin</mi>\u0000 <mi>c</mi>\u0000 </msup>\u0000 <annotation>${rm spin}^c$</annotation>\u0000 </semantics></math> structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3-manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov-type inequalities that relate topological quantities of 4-manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4-manifolds, and nonsmoothable unoriented surfaces in 4-manifolds.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the tau invariants in instanton and monopole Floer theories","authors":"Sudipta Ghosh,&nbsp;Zhenkun Li,&nbsp;C.-M. Michael Wong","doi":"10.1112/topo.12346","DOIUrl":"https://doi.org/10.1112/topo.12346","url":null,"abstract":"<p>We unify two existing approaches to the <i>tau</i> invariants in instanton and monopole Floer theories, by identifying <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>τ</mi>\u0000 <mi>G</mi>\u0000 </msub>\u0000 <annotation>$tau _{mathrm{G}}$</annotation>\u0000 </semantics></math>, defined by the second author via the <i>minus</i> flavors <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <munder>\u0000 <mo>KHI</mo>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <annotation>$underline{operatorname{KHI}}^-$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <munder>\u0000 <mo>KHM</mo>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <annotation>$underline{operatorname{KHM}}^-$</annotation>\u0000 </semantics></math> of the knot homologies, with <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>τ</mi>\u0000 <mi>G</mi>\u0000 <mo>♯</mo>\u0000 </msubsup>\u0000 <annotation>$tau ^sharp _{mathrm{G}}$</annotation>\u0000 </semantics></math>, defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <munder>\u0000 <mo>KHI</mo>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <annotation>$underline{operatorname{KHI}}^-$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <munder>\u0000 <mo>KHM</mo>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <annotation>$underline{operatorname{KHM}}^-$</annotation>\u0000 </semantics></math> for twist knots.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12346","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Brieskorn spheres, cyclic group actions and the Milnor conjecture","authors":"David Baraglia,&nbsp;Pedram Hekmati","doi":"10.1112/topo.12339","DOIUrl":"https://doi.org/10.1112/topo.12339","url":null,"abstract":"&lt;p&gt;In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;θ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$theta ^{(c)}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; defined by the first author satisfy &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;θ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$theta ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for torus knots, whenever &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;annotation&gt;$c$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a prime not dividing &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ab$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Since &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;θ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$theta ^{(c)}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a lower bound for","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12339","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum: Étale cohomology, purity and formality with torsion coefficients","authors":"Joana Cirici,&nbsp;Geoffroy Horel","doi":"10.1112/topo.12348","DOIUrl":"https://doi.org/10.1112/topo.12348","url":null,"abstract":"<p>Proposition 6.9 in (J. Topol. <b>15</b> (2022), no. 4, 2270–2297) is incorrect without a connectivity assumption. In this note, we provide a counter-example, give a correct proof of the modified proposition and explain the other changes that need to be made to [1].</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12348","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}