{"title":"Derived deformation theory of crepant curves","authors":"Gavin Brown, Michael Wemyss","doi":"10.1112/topo.12359","DOIUrl":"https://doi.org/10.1112/topo.12359","url":null,"abstract":"<p>This paper determines the full, derived deformation theory of certain smooth rational curves <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathrm{C}$</annotation>\u0000 </semantics></math> in Calabi–Yau 3-folds, by determining all higher <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <annotation>$mathrm{A}_infty$</annotation>\u0000 </semantics></math>-products in its controlling DG-algebra. This geometric setup includes very general cases where <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathrm{C}$</annotation>\u0000 </semantics></math> does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3-fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the non-commutative deformation theory of <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathrm{C}$</annotation>\u0000 </semantics></math> is described via a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari (<i>Adv. Theor. Math. Phys</i>. <b>7</b> (2003) 619–665), Aspinwall–Katz (<i>Comm. Math. Phys</i>.. <b>264</b> (2006) 227–253) and Curto–Morrison (<i>J. Algebraic Geom</i>. <b>22</b> (2013) 599–627). Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3-folds, which lift Artin's (<i>Amer. J. Math</i>. <b>84</b> (1962) 485–496) celebrated results from surfaces.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12359","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142438990","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":"Calabi–Yau structures on Rabinowitz Fukaya categories","authors":"Hanwool Bae, Wonbo Jeong, Jongmyeong Kim","doi":"10.1112/topo.12361","DOIUrl":"https://doi.org/10.1112/topo.12361","url":null,"abstract":"<p>In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> of dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$2n$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(n-1)$</annotation>\u0000 </semantics></math>-Calabi–Yau, assuming that the wrapped Fukaya category of <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> admits an at most countable set of Lagrangians that generate it and satisfy some finiteness condition on morphism spaces between them.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434974","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":"Oriented Birkhoff sections of Anosov flows","authors":"Masayuki Asaoka, Christian Bonatti, Théo Marty","doi":"10.1112/topo.12356","DOIUrl":"https://doi.org/10.1112/topo.12356","url":null,"abstract":"<p>This paper gives three different proofs (independently obtained by the three authors) of the following fact: given an Anosov flow on an oriented 3-manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-covered positively twisted.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434973","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 homology of big mapping class groups","authors":"Martin Palmer, Xiaolei Wu","doi":"10.1112/topo.12358","DOIUrl":"https://doi.org/10.1112/topo.12358","url":null,"abstract":"<p>We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of <i>dissipated groups</i>; the other is a general homological stability result for mapping class groups of infinite-type surfaces.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12358","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435193","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 realisation result for moduli spaces of group actions on the line","authors":"Joaquín Brum, Nicolás Matte Bon, Cristóbal Rivas, Michele Triestino","doi":"10.1112/topo.12357","DOIUrl":"https://doi.org/10.1112/topo.12357","url":null,"abstract":"<p>Given a finitely generated group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, the possible actions of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>Φ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Y, Phi)$</annotation>\u0000 </semantics></math> naturally associated with <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and uniquely defined up to flow equivalence, that we call the <i>Deroin space</i> of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>. We show a realisation result: every expansive flow <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>Φ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Y, Phi)$</annotation>\u0000 </semantics></math> on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435192","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":"Morse numbers of complex polynomials","authors":"Laurenţiu Maxim, Mihai Tibăr","doi":"10.1112/topo.12362","DOIUrl":"https://doi.org/10.1112/topo.12362","url":null,"abstract":"<p>To a complex polynomial function <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> with arbitrary singularities, we associate the number of Morse points in a general linear Morsification <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>:</mo>\u0000 <mo>=</mo>\u0000 <mi>f</mi>\u0000 <mo>−</mo>\u0000 <mi>t</mi>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 <annotation>$f_{t}:= f - tell$</annotation>\u0000 </semantics></math>. We produce computable algebraic formulae in terms of invariants of <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> for the numbers of stratwise Morse trajectories that abut, as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$trightarrow 0$</annotation>\u0000 </semantics></math>, to some point of the space or at infinity.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12362","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435191","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":"Stated SL(\u0000 \u0000 n\u0000 $n$\u0000 )-skein modules and algebras","authors":"Thang T. Q. Lê, Adam S. Sikora","doi":"10.1112/topo.12350","DOIUrl":"https://doi.org/10.1112/topo.12350","url":null,"abstract":"<p>We develop a theory of stated SL(<span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>)-skein modules, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_n(M,mathcal {N})$</annotation>\u0000 </semantics></math>, of 3-manifolds <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> marked with intervals <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$mathcal {N}$</annotation>\u0000 </semantics></math> in their boundaries. These skein modules, generalizing stated SL(2)-modules of the first author, stated SL(3)-modules of Higgins', and SU(n)-skein modules of the second author, consist of linear combinations of framed, oriented graphs, called <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-webs, with ends in <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$mathcal {N}$</annotation>\u0000 </semantics></math>, considered up to skein relations of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>s</mi>\u0000 <msub>\u0000 <mi>l</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$U_q(sl_n)$</annotation>\u0000 </semantics></math>-Reshetikhin–Turaev functor on tangles, involving coupons representing the anti-symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3-manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> along a disk resulting in a 3-manifold <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>M</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <annotation>$M^{prime }$</annotation>\u0000 </semantics></math> yields a homomorphism <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142041631","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}
Jason Behrstock, Mark Hagen, Alexandre Martin, Alessandro Sisto
{"title":"A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups","authors":"Jason Behrstock, Mark Hagen, Alexandre Martin, Alessandro Sisto","doi":"10.1112/topo.12351","DOIUrl":"https://doi.org/10.1112/topo.12351","url":null,"abstract":"<p>We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967845","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":"Regularity of limit sets of Anosov representations","authors":"Tengren Zhang, Andrew Zimmer","doi":"10.1112/topo.12355","DOIUrl":"https://doi.org/10.1112/topo.12355","url":null,"abstract":"<p>In this paper, we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>α</mi>\u0000 </msup>\u0000 <annotation>$C^{alpha }$</annotation>\u0000 </semantics></math>-submanifold of the real projective space for some <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$alpha in (1,2)$</annotation>\u0000 </semantics></math>. We also calculate the optimal value of <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> in terms of the eigenvalue data of the Anosov representation.</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":"141967139","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":"Coarse cubical rigidity","authors":"Elia Fioravanti, Ivan Levcovitz, 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}
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