Journal of Topology最新文献

{"title":"Gromov–Witten theory of complete intersections via nodal invariants","authors":"Hülya Argüz,&nbsp;Pierrick Bousseau,&nbsp;Rahul Pandharipande,&nbsp;Dimitri Zvonkine","doi":"10.1112/topo.12284","DOIUrl":"10.1112/topo.12284","url":null,"abstract":"<p>We provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"264-343"},"PeriodicalIF":1.1,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12284","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47334185","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":"Families of diffeomorphisms and concordances detected by trivalent graphs","authors":"Boris Botvinnik,&nbsp;Tadayuki Watanabe","doi":"10.1112/topo.12283","DOIUrl":"10.1112/topo.12283","url":null,"abstract":"&lt;p&gt;We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Diff&lt;/mi&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&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;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$pi _*(Bmathrm{Diff}_{partial }(D^d))otimes {mathbb {Q}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are lifted to homotopy groups of the moduli space of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;h&lt;/mi&gt;\u0000 &lt;annotation&gt;$h$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-cobordisms &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Diff&lt;/mi&gt;\u0000 &lt;mo&gt;⊔&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;I&lt;/mi&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;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$pi _*(Bmathrm{Diff}_{sqcup }(D^dtimes I))otimes {mathbb {Q}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. As a geometrical application, we show that those elements in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Diff&lt;/mi&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"207-233"},"PeriodicalIF":1.1,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41633462","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":"Simplicial volume and essentiality of manifolds fibered over spheres","authors":"Thorben Kastenholz,&nbsp;Jens Reinhold","doi":"10.1112/topo.12286","DOIUrl":"10.1112/topo.12286","url":null,"abstract":"<p>We study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>⩾</mo>\u0000 <mn>7</mn>\u0000 </mrow>\u0000 <annotation>$2n +1 geqslant 7$</annotation>\u0000 </semantics></math> with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 2$</annotation>\u0000 </semantics></math>: we prove that their total spaces are rationally inessential if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 3$</annotation>\u0000 </semantics></math>, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"192-206"},"PeriodicalIF":1.1,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12286","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41675764","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 Milnor number of non-isolated singularities of holomorphic foliations and its topological invariance","authors":"Arturo Fernández-Pérez,&nbsp;Gilcione Nonato Costa,&nbsp;Rudy Rosas Bazán","doi":"10.1112/topo.12281","DOIUrl":"10.1112/topo.12281","url":null,"abstract":"<p>We define the Milnor number of a one-dimensional holomorphic foliation <math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> as the intersection number of two holomorphic sections with respect to a compact connected component <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> of its singular set. Under certain conditions, we prove that the Milnor number of <math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> on a three-dimensional manifold with respect to <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> is invariant by <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$C^1$</annotation>\u0000 </semantics></math> topological equivalences.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"176-191"},"PeriodicalIF":1.1,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41519115","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":"Large genus asymptotics for lengths of separating closed geodesics on random surfaces","authors":"Xin Nie,&nbsp;Yunhui Wu,&nbsp;Yuhao Xue","doi":"10.1112/topo.12276","DOIUrl":"10.1112/topo.12276","url":null,"abstract":"<p>In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus <math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math> with respect to the Weil–Petersson measure on the moduli space <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <annotation>$mathcal {M}_g$</annotation>\u0000 </semantics></math>. We show that as <math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math> goes to infinity, a generic surface <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>∈</mo>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$Xin mathcal {M}_g$</annotation>\u0000 </semantics></math> satisfies asymptotically: \u0000\u0000 </p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"106-175"},"PeriodicalIF":1.1,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377726","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":"End-periodic homeomorphisms and volumes of mapping tori","authors":"Elizabeth Field,&nbsp;Heejoung Kim,&nbsp;Christopher Leininger,&nbsp;Marissa Loving","doi":"10.1112/topo.12277","DOIUrl":"https://doi.org/10.1112/topo.12277","url":null,"abstract":"<p>Given an irreducible, end-periodic homeomorphism <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <mi>S</mi>\u0000 <mo>→</mo>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation>$f: S rightarrow S$</annotation>\u0000 </semantics></math> of a surface with finitely many ends, all accumulated by genus, the mapping torus, <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>f</mi>\u0000 </msub>\u0000 <annotation>$M_f$</annotation>\u0000 </semantics></math>, is the interior of a compact, irreducible, atoroidal 3-manifold <math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>M</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <mi>f</mi>\u0000 </msub>\u0000 <annotation>$overline{M}_f$</annotation>\u0000 </semantics></math> with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>M</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <mi>f</mi>\u0000 </msub>\u0000 <annotation>$overline{M}_f$</annotation>\u0000 </semantics></math> in terms of the translation length of <math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> on the pants graph of <math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"57-105"},"PeriodicalIF":1.1,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12277","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50122479","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":"Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groups","authors":"Eduard Schesler,&nbsp;Matthew C. B. Zaremsky","doi":"10.1112/topo.12278","DOIUrl":"10.1112/topo.12278","url":null,"abstract":"<p>The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to <math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>$mathbb {Z}$</annotation>\u0000 </semantics></math> whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type <math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>F</mo>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$operatorname{F}_2$</annotation>\u0000 </semantics></math> but not <math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>FP</mo>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <annotation>$operatorname{FP}_3$</annotation>\u0000 </semantics></math>, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz–Norin–Wise involving Bestvina–Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"20-56"},"PeriodicalIF":1.1,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48971862","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":"A criterion for density of the isoperiodic leaves in rank one affine invariant orbifolds","authors":"Florent Ygouf","doi":"10.1112/topo.12279","DOIUrl":"https://doi.org/10.1112/topo.12279","url":null,"abstract":"<p>We define on any affine invariant orbifold <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> a foliation <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>F</mi>\u0000 <mi>M</mi>\u0000 </msup>\u0000 <annotation>$mathcal {F}^{mathcal {M}}$</annotation>\u0000 </semantics></math> that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification of the dynamical behavior of the leaves of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>F</mi>\u0000 <mi>M</mi>\u0000 </msup>\u0000 <annotation>$mathcal {F}^{mathcal {M}}$</annotation>\u0000 </semantics></math> when <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> is a connected component of a Prym eigenform locus in genus 2 or 3 and the second provides the first examples of dense isoperiodic leaves in the stratum <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {H}(2,1,1)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":"1-19"},"PeriodicalIF":1.1,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12279","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50146539","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 criterion for density of the isoperiodic leaves in rank one affine invariant orbifolds","authors":"Florent Ygouf","doi":"10.1112/topo.12279","DOIUrl":"https://doi.org/10.1112/topo.12279","url":null,"abstract":"We define on any affine invariant orbifold M$mathcal {M}$ a foliation FM$mathcal {F}^{mathcal {M}}$ that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification of the dynamical behavior of the leaves of FM$mathcal {F}^{mathcal {M}}$ when M$mathcal {M}$ is a connected component of a Prym eigenform locus in genus 2 or 3 and the second provides the first examples of dense isoperiodic leaves in the stratum H(2,1,1)$mathcal {H}(2,1,1)$ .","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63413602","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":"The correspondence induced on the pillowcase by the earring tangle","authors":"Guillem Cazassus,&nbsp;Christopher Herald,&nbsp;Paul Kirk,&nbsp;Artem Kotelskiy","doi":"10.1112/topo.12272","DOIUrl":"10.1112/topo.12272","url":null,"abstract":"<p>The earring tangle consists of four strands <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 <mtext>pt</mtext>\u0000 <mo>×</mo>\u0000 <mi>I</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <mi>I</mi>\u0000 </mrow>\u0000 <annotation>$4text{pt} times I subset S^2 times I$</annotation>\u0000 </semantics></math> and one meridian around one of the strands. Equipping this tangle with a nontrivial <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$SO(3)$</annotation>\u0000 </semantics></math> bundle, we show that its traceless <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$SU(2)$</annotation>\u0000 </semantics></math> flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"15 4","pages":"2472-2543"},"PeriodicalIF":1.1,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47353465","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}