{"title":"The Mumford conjecture (after Bianchi)","authors":"Ronno Das, 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":"Groups with exotic finiteness properties from complex Morse theory","authors":"Claudio Llosa Isenrich, Pierre Py","doi":"10.1112/topo.70013","DOIUrl":"https://doi.org/10.1112/topo.70013","url":null,"abstract":"<p>Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>, new hyperbolic groups admitting surjective homomorphisms to <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>${mathbb {Z}}$</annotation>\u0000 </semantics></math> and to <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>${mathbb {Z}}^{2}$</annotation>\u0000 </semantics></math>, whose kernel is of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <annotation>$mathcal {F}_{k}$</annotation>\u0000 </semantics></math> but not of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mathcal {F}_{k+1}$</annotation>\u0000 </semantics></math>. By a fibre product construction, we also find examples of non-normal subgroups of Kähler groups with exotic finiteness properties.</p>","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":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143513792","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}
Agnès Beaudry, Michael A. Hill, Tyler Lawson, XiaoLin Danny Shi, Mingcong Zeng
{"title":"On the slice spectral sequence for quotients of norms of Real bordism","authors":"Agnès Beaudry, Michael A. Hill, Tyler Lawson, XiaoLin Danny Shi, Mingcong Zeng","doi":"10.1112/topo.70015","DOIUrl":"https://doi.org/10.1112/topo.70015","url":null,"abstract":"<p>In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <msup>\u0000 <mi>U</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 </msup>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mspace></mspace>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$MU^{(!(C_{2^n})!)}$</annotation>\u0000 </semantics></math> by permutation summands. These quotients are of interest because of their close relationship with higher real <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>B</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 </msup>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mspace></mspace>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$BP^{(!(C_{2^n})!)}langle m,mrangle$</annotation>\u0000 </semantics></math>. These spectra serve as natural equivariant generalizations of connective integral Morava <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theories. We provide a complete computation of the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>σ</mi>\u0000 </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}
{"title":"Milnor fiber consistency via flatness","authors":"Alex Hof","doi":"10.1112/topo.70014","DOIUrl":"https://doi.org/10.1112/topo.70014","url":null,"abstract":"<p>We describe a new algebro-geometric perspective on the study of the Milnor fibration and, as a first step toward putting it into practice, prove powerful criteria for a deformation of a holomorphic function germ to admit a stratification on its domain partially satisfying the Thom condition and, more generally, to respect the Milnor fibration of the original germ in an appropriate sense. As corollaries, we obtain a method of partitioning the space of homogeneous polynomials of a fixed degree into finitely many locally closed subsets such that the fiber diffeomorphism type of the Milnor fibration is constant along each subset and a criterion under which deformations of a function with critical locus a complete intersection will be well-behaved.</p>","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":"143513791","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":"Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces","authors":"Benjamin Gammage, Maxim Jeffs","doi":"10.1112/topo.70012","DOIUrl":"https://doi.org/10.1112/topo.70012","url":null,"abstract":"<p>We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has, in fact, two natural mirrors, one of which is a derived scheme. These two mirrors are related via a nongeometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143121180","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":"Neck-pinching of \u0000 \u0000 \u0000 C\u0000 \u0000 P\u0000 1\u0000 \u0000 \u0000 $mathbb {C}{rm P}^1$\u0000 -structures in the \u0000 \u0000 \u0000 \u0000 PSL\u0000 2\u0000 \u0000 C\u0000 \u0000 ${rm PSL}_2mathbb {C}$\u0000 -character variety","authors":"Shinpei Baba","doi":"10.1112/topo.70010","DOIUrl":"10.1112/topo.70010","url":null,"abstract":"<p>We characterize a certain neck-pinching degeneration of (marked) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}{rm P}^1$</annotation>\u0000 </semantics></math>-structures on a closed oriented surface <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> of genus at least two. In a more general setting, we take a path of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}{rm P}^1$</annotation>\u0000 </semantics></math>-structures <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>⩾</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$C_t nobreakspace (t geqslant 0)$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> that leaves every compact subset in its deformation space, such that the holonomy of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <annotation>$C_t$</annotation>\u0000 </semantics></math> converges in the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>PSL</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation>${rm PSL}_2mathbb {C}$</annotation>\u0000 </semantics></math>-character variety as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$t rightarrow infty$</annotation>\u0000 </semantics></math>. Then, it is well known that the complex structure <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>X</m","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11685183/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142916395","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":"Monopoles and Landau–Ginzburg models III: A gluing theorem","authors":"Donghao Wang","doi":"10.1112/topo.12360","DOIUrl":"https://doi.org/10.1112/topo.12360","url":null,"abstract":"<p>This is the third paper of this series. In Wang [Monopoles and Landau-Ginzburg models II: Floer homology. arXiv:2005.04333, 2020], we defined the monopole Floer homology for any pair <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Y,omega)$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> is a compact oriented 3-manifold with toroidal boundary and <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> is a suitable closed 2-form viewed as a decoration. In this paper, we establish a gluing theorem for this Floer homology when two such 3-manifolds are glued suitably along their common boundary, assuming that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>∂</mi>\u0000 <mi>Y</mi>\u0000 </mrow>\u0000 <annotation>$partial Y$</annotation>\u0000 </semantics></math> is disconnected, and <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> is small and yet non-vanishing on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>∂</mi>\u0000 <mi>Y</mi>\u0000 </mrow>\u0000 <annotation>$partial Y$</annotation>\u0000 </semantics></math>. As applications, we construct a monopole Floer 2-functor and the generalized cobordism maps. Using results of Kronheimer–Mrowka and Ni, it is shown that for any such 3-manifold <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> that is irreducible, this Floer homology detects the Thurston norm on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>∂</mi>\u0000 <mi>Y</mi>\u0000 <mo>;</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_2(Y,partial Y;mathbb {R})$</annotation>\u0000 </semantics></math> and the fiberness of <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math>. Finally, we show that our construction recovers the monopole link Floer homology f","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143119998","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":"Nonabelian basechange theorems and étale homotopy theory","authors":"Peter J. Haine, Tim Holzschuh, Sebastian Wolf","doi":"10.1112/topo.70009","DOIUrl":"https://doi.org/10.1112/topo.70009","url":null,"abstract":"<p>This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in étale cohomology (i.e., prove analogues of the classical statements for sheaves of <i>spaces</i>). Second, we apply these theorems to prove a number of results about the étale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber–Gabber affine analogue of the proper basechange theorem, and Fujiwara–Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt–Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite étale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of Künneth formulas for the étale homotopy type.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142762241","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}
Irving Dai, Matthew Hedden, Abhishek Mallick, Matthew Stoffregen
{"title":"Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology","authors":"Irving Dai, Matthew Hedden, Abhishek Mallick, Matthew Stoffregen","doi":"10.1112/topo.70008","DOIUrl":"https://doi.org/10.1112/topo.70008","url":null,"abstract":"<p>We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinzón-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns that act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annotation>$tau$</annotation>\u0000 </semantics></math>-invariant. Our results also recover and extend several theorems in this area established using instanton Floer homology.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737409","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}
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