Journal of Topology最新文献

{"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}

{"title":"On the Smith–Thom deficiency of Hilbert squares","authors":"Viatcheslav Kharlamov,&nbsp;Rareş Răsdeaconu","doi":"10.1112/topo.12345","DOIUrl":"https://doi.org/10.1112/topo.12345","url":null,"abstract":"<p>We give an expression for the Smith–Thom deficiency of the Hilbert square <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>X</mi>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>2</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$X^{[2]}$</annotation>\u0000 </semantics></math> of a smooth real algebraic variety <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>X</mi>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>2</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$X^{[2]}$</annotation>\u0000 </semantics></math> in the case of projective complete intersections, and show that with a few exceptions, no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12345","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141182129","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":"Invertible topological field theories","authors":"Christopher Schommer-Pries","doi":"10.1112/topo.12335","DOIUrl":"https://doi.org/10.1112/topo.12335","url":null,"abstract":"&lt;p&gt;A &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(infty,n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-category of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-bordisms (embedded into &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {R}^infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and equipped with a tangential &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;ξ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(X,xi)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-structure) that lands in the Picard subcategory of the target symmetric monoidal &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(infty,n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-category. We classify these field theories in terms of the cohomology of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(n-d)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(infty,n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-category of bordisms with &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;Ω&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;ξ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141164808","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":"Non-accessible localizations","authors":"J. Daniel Christensen","doi":"10.1112/topo.12336","DOIUrl":"https://doi.org/10.1112/topo.12336","url":null,"abstract":"<p>In a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$mathcal {U}$</annotation>\u0000 </semantics></math>. When specialized to an appropriate family, this produces a localization which when interpreted in the <span></span><math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-topos of spaces agrees with the localization corresponding to <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math>. Our approach generalizes the approach of Casacuberta et al. (Adv. Math. <b>197</b> (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any <span></span><math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-topos. Second, while the local objects produced by Casacuberta et al. are always 1-types, our construction can produce <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-types, for any <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>. This is new, even in the <span></span><math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12336","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085022","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}