Journal of Topology最新文献

{"title":"Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory","authors":"Markus Upmeier","doi":"10.1112/topo.70025","DOIUrl":"https://doi.org/10.1112/topo.70025","url":null,"abstract":"<p>We develop a general theory of pushforward operations for principal <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>-bundles equipped with a certain type of orientation. In the case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mi>B</mi>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$G={Bmathrm{U}(1)}$</annotation>\u0000 </semantics></math> and orientations in twisted K-theory, we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation. As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>B</mi>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${Bmathrm{U}(1)}$</annotation>\u0000 </semantics></math>-action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144171596","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":"Bounded projections to the \u0000 \u0000 Z\u0000 $mathcal {Z}$\u0000 -factor graph","authors":"Matt Clay,&nbsp;Caglar Uyanik","doi":"10.1112/topo.70024","DOIUrl":"https://doi.org/10.1112/topo.70024","url":null,"abstract":"&lt;p&gt;Suppose that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a free product &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;mi&gt;⋯&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$G = A_1 * A_2* cdots * A_k * F_N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where each of the groups &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$A_i$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is torsion-free and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$F_N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a free group of rank &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;annotation&gt;$N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {O}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {O}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where a given element has bounded length to the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {Z}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as a hyperbolic group relative to the collection of subgroups &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140692","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":"Simple closed curves, non-kernel homology and Magnus embedding","authors":"Adam Klukowski","doi":"10.1112/topo.70023","DOIUrl":"https://doi.org/10.1112/topo.70023","url":null,"abstract":"<p>We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite solvable quotient of the fundamental group we exhibit a cover whose homology is not generated by the lifts of curves in the complement of its kernel. We explain how the existing approach of Malestein and Putman (for branched covers) relates to the Magnus embedding, and by doing so we simplify their construction. We then generalise it to unbranched covers by producing embeddings of surface groups into units of certain graded associative algebras, which may be of independent interest.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143893027","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: Strong 𝔸1-invariance of 𝔸1-connected components of reductive algebraic groups","authors":"Chetan Balwe,&nbsp;Amit Hogadi,&nbsp;Anand Sawant","doi":"10.1112/topo.70022","DOIUrl":"https://doi.org/10.1112/topo.70022","url":null,"abstract":"<p>The proof of [2, Lemma 5.1] is incomplete as it relies on some results in [4], the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of [2, Lemma 5.1].</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143840613","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":"The Picard group in equivariant homotopy theory via stable module categories","authors":"Achim Krause","doi":"10.1112/topo.70020","DOIUrl":"https://doi.org/10.1112/topo.70020","url":null,"abstract":"<p>We develop a mechanism of “isotropy separation for compact objects” that explicitly describes an invertible <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of possible combinations of geometric fixed points of invertible <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>-spectra in the case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mn>5</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$G=A_5$</annotation>\u0000 </semantics></math>. A further application is given by showing that the Picard groups of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mo>Sp</mo>\u0000 <mi>G</mi>\u0000 </msup>\u0000 <annotation>$operatorname{Sp}^G$</annotation>\u0000 </semantics></math> and a category of derived Mackey functors agree.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143801878","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":"Structure of quasiconvex virtual joins","authors":"Lawk Mineh","doi":"10.1112/topo.70021","DOIUrl":"https://doi.org/10.1112/topo.70021","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a relatively hyperbolic group and let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;annotation&gt;$Q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;$R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups &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;Q&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Q^{prime } leqslant _f Q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &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;R&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$R^{prime } leqslant _f R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that the subgroup join &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;msup&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;⟩&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$langle Q^{prime }, R^{prime } rangle$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is also relatively quasiconvex, given suitable assumptions on the profinite topology of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We show that the intersections of such joins with maximal parabolic subgroups of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are themselves joins of intersections of the factor subgroups &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$Q^{prime }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/mat","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143778397","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":"A classification of infinite staircases for Hirzebruch surfaces","authors":"Nicki Magill,&nbsp;Ana Rita Pires,&nbsp;Morgan Weiler","doi":"10.1112/topo.70017","DOIUrl":"https://doi.org/10.1112/topo.70017","url":null,"abstract":"<p>The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four-ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an “infinite staircase,” that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair-step pattern. Later work of Usher and Cristofaro-Gardiner–Holm–Mandini–Pires suggested that while four-dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly nongeneric. This paper concludes the systematic study of one-point blowups of the complex projective plane, building on previous work of Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff, Magill-McDuff-Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro-Gardiner–Holm–Mandini–Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$1/3$</annotation>\u0000 </semantics></math>. We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres).</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581764","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 parameterized Tate construction","authors":"J. D. Quigley,&nbsp;Jay Shah","doi":"10.1112/topo.70018","DOIUrl":"https://doi.org/10.1112/topo.70018","url":null,"abstract":"&lt;p&gt;We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;̂&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$widehat{G}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of a finite group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; by a compact Lie group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, which we call the parameterized Tate construction &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$(-)^{t_G K}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Our main theorem establishes the coincidence of three conceptually distinct approaches to its construction when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is also finite: one via recollement theory for the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-free &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;̂&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$widehat{G}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-family, another via parameterized ambidexterity for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-local systems, and the last via parameterized assembly maps. We also show that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$(-)^{t_G K}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; uniquely admits the structure o","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143564680","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":"The Mumford conjecture (after Bianchi)","authors":"Ronno Das,&nbsp;Dan Petersen","doi":"10.1112/topo.70016","DOIUrl":"https://doi.org/10.1112/topo.70016","url":null,"abstract":"<p>We give a self-contained and streamlined rendition of Andrea Bianchi's recent proof of the Mumford conjecture using moduli spaces of branched covers.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143521943","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 slice spectral sequence for quotients of norms of Real bordism","authors":"Agnès Beaudry,&nbsp;Michael A. Hill,&nbsp;Tyler Lawson,&nbsp;XiaoLin Danny Shi,&nbsp;Mingcong Zeng","doi":"10.1112/topo.70015","DOIUrl":"https://doi.org/10.1112/topo.70015","url":null,"abstract":"&lt;p&gt;In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$MU^{(!(C_{2^n})!)}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; by permutation summands. These quotients are of interest because of their close relationship with higher real &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mspace&gt;&lt;/mspace&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;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;⟩&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$BP^{(!(C_{2^n})!)}langle m,mrangle$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. These spectra serve as natural equivariant generalizations of connective integral Morava &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-theories. We provide a complete computation of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;σ&lt;/mi&gt;\u0000 &lt;/msub","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143521902","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}