We consider the question of extending a smooth homotopy coherent finite cyclic group action on the boundary of a smooth 4-manifold to its interior. As a result, we prove that Dehn twists along any Seifert homology sphere, except the 3-sphere, on their simply connected positive-definite fillings are infinite order exotic.
{"title":"Exotic Dehn twists and homotopy coherent group actions","authors":"Sungkyung Kang, JungHwan Park, Masaki Taniguchi","doi":"arxiv-2409.11806","DOIUrl":"https://doi.org/arxiv-2409.11806","url":null,"abstract":"We consider the question of extending a smooth homotopy coherent finite\u0000cyclic group action on the boundary of a smooth 4-manifold to its interior. As\u0000a result, we prove that Dehn twists along any Seifert homology sphere, except\u0000the 3-sphere, on their simply connected positive-definite fillings are infinite\u0000order exotic.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of $infty$-categories of truncated right-modules over a unital $infty$-operad $mathcal{O}$. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as $mathcal{O}$ varies, and generalise these results to the level of Morita $(infty,2)$-categories. Applied to the ${rm BO}(d)$-framed $E_d$-operad, this extends Goodwillie-Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the $E_d$-operad, it yields new versions of embedding calculus, such as one for topological embeddings, based on ${rm BTop}(d)$, or one similar to Boavida de Brito-Weiss' configuration categories, based on ${rm BAut}(E_d)$. In addition, we prove a delooping result in the context of embedding calculus, establish a convergence result for topological embedding calculus, improve upon the smooth convergence result of Goodwillie, Klein, and Weiss, and deduce an Alexander trick for homology 4-spheres.
{"title":"$infty$-operadic foundations for embedding calculus","authors":"Manuel Krannich, Alexander Kupers","doi":"arxiv-2409.10991","DOIUrl":"https://doi.org/arxiv-2409.10991","url":null,"abstract":"Motivated by applications to spaces of embeddings and automorphisms of\u0000manifolds, we consider a tower of $infty$-categories of truncated\u0000right-modules over a unital $infty$-operad $mathcal{O}$. We study monoidality\u0000and naturality properties of this tower, identify its layers, describe the\u0000difference between the towers as $mathcal{O}$ varies, and generalise these\u0000results to the level of Morita $(infty,2)$-categories. Applied to the ${rm\u0000BO}(d)$-framed $E_d$-operad, this extends Goodwillie-Weiss' embedding calculus\u0000and its layer identification to the level of bordism categories. Applied to\u0000other variants of the $E_d$-operad, it yields new versions of embedding\u0000calculus, such as one for topological embeddings, based on ${rm BTop}(d)$, or\u0000one similar to Boavida de Brito-Weiss' configuration categories, based on ${rm\u0000BAut}(E_d)$. In addition, we prove a delooping result in the context of\u0000embedding calculus, establish a convergence result for topological embedding\u0000calculus, improve upon the smooth convergence result of Goodwillie, Klein, and\u0000Weiss, and deduce an Alexander trick for homology 4-spheres.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We fully describe all horocycle orbit closures in $ mathbb{Z} $-covers of compact hyperbolic surfaces. Our results rely on a careful analysis of the efficiency of all distance minimizing geodesic rays in the cover. As a corollary we obtain in this setting that all non-maximal horocycle orbit closures, while fractal, have integer Hausdorff dimension.
{"title":"Classification of horocycle orbit closures in $ mathbb{Z} $-covers","authors":"James Farre, Or Landesberg, Yair Minsky","doi":"arxiv-2409.10004","DOIUrl":"https://doi.org/arxiv-2409.10004","url":null,"abstract":"We fully describe all horocycle orbit closures in $ mathbb{Z} $-covers of\u0000compact hyperbolic surfaces. Our results rely on a careful analysis of the\u0000efficiency of all distance minimizing geodesic rays in the cover. As a\u0000corollary we obtain in this setting that all non-maximal horocycle orbit\u0000closures, while fractal, have integer Hausdorff dimension.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this note is to share the observation that the set of elementary operations of Turing on lattice knots can be reduced to just one type of simple local switches.
本论文旨在与大家分享一个观察结果，即图灵在网格结上的基本操作集可以简化为一种简单局部开关。
{"title":"A note on lattice knots","authors":"Sasha Anan'in, Alexandre Grishkov, Dmitrii Korshunov","doi":"arxiv-2409.10691","DOIUrl":"https://doi.org/arxiv-2409.10691","url":null,"abstract":"The aim of this note is to share the observation that the set of elementary\u0000operations of Turing on lattice knots can be reduced to just one type of simple\u0000local switches.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Lambda$ be a subfamily of the moduli space of degree $Dge2$ polynomials defined by a finite number of parabolic relations. Let $Omega$ be a bounded stable component of $Lambda$ with the property that all critical points are attracted by either the persistent parabolic cycles or by attracting cycles in $mathbb C$. We construct a positive semi-definite pressure form on $Omega$ and show that it defines a path metric on $Omega$. This provides a counterpart in complex dynamics of the pressure metric on cusped Hitchin components recently studied by Kao and Bray-Canary-Kao-Martone.
{"title":"Pressure path metrics on parabolic families of polynomials","authors":"Fabrizio Bianchi, Yan Mary He","doi":"arxiv-2409.10462","DOIUrl":"https://doi.org/arxiv-2409.10462","url":null,"abstract":"Let $Lambda$ be a subfamily of the moduli space of degree $Dge2$\u0000polynomials defined by a finite number of parabolic relations. Let $Omega$ be\u0000a bounded stable component of $Lambda$ with the property that all critical\u0000points are attracted by either the persistent parabolic cycles or by attracting\u0000cycles in $mathbb C$. We construct a positive semi-definite pressure form on\u0000$Omega$ and show that it defines a path metric on $Omega$. This provides a\u0000counterpart in complex dynamics of the pressure metric on cusped Hitchin\u0000components recently studied by Kao and Bray-Canary-Kao-Martone.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a generalization of the Bers' simultaneous uniformization theorem in the world of algebraic correspondences. More precisely, we construct algebraic correspondences that simultaneously uniformize a pair of non-homeomorphic genus zero orbifolds. We also present a complex-analytic realization of the Teichm"uller space of a punctured sphere in the space of correspondences.
{"title":"Simultaneous Uniformization and Algebraic Correspondences","authors":"Mahan Mj, Sabyasachi Mukherjee","doi":"arxiv-2409.10468","DOIUrl":"https://doi.org/arxiv-2409.10468","url":null,"abstract":"We prove a generalization of the Bers' simultaneous uniformization theorem in\u0000the world of algebraic correspondences. More precisely, we construct algebraic\u0000correspondences that simultaneously uniformize a pair of non-homeomorphic genus\u0000zero orbifolds. We also present a complex-analytic realization of the\u0000Teichm\"uller space of a punctured sphere in the space of correspondences.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"202 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A closed 3-manifold $M$ may be described up to some indeterminacy by a Heegaard diagram $mathcal{D}$. The question "Does $M$ smoothly embed in $mathbb{R}^4$?'' is equivalent to a property of $mathcal{D}$ which we call $textit{doubly unlinked}$ (DU). This perspective leads to an enhancement of Hantzsche's embedding obstruction.
{"title":"Enhanced Hantzsche Theorem","authors":"Michael H. Freedman","doi":"arxiv-2409.09983","DOIUrl":"https://doi.org/arxiv-2409.09983","url":null,"abstract":"A closed 3-manifold $M$ may be described up to some indeterminacy by a\u0000Heegaard diagram $mathcal{D}$. The question \"Does $M$ smoothly embed in\u0000$mathbb{R}^4$?'' is equivalent to a property of $mathcal{D}$ which we call\u0000$textit{doubly unlinked}$ (DU). This perspective leads to an enhancement of\u0000Hantzsche's embedding obstruction.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{"u}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichm{"u}ller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting.
{"title":"The horocyclic metric on Teichm{ü}ller spaces","authors":"Hideki MiyachiIRMA, Ken'Ichi OhshikaIRMA, Athanase PapadopoulosIRMA","doi":"arxiv-2409.10082","DOIUrl":"https://doi.org/arxiv-2409.10082","url":null,"abstract":"In his paper Minimal stretch maps between hyperbolic surfaces, William\u0000Thurston defined a norm on the tangent space to Teichm{\"u}ller space of a\u0000hyperbolic surface, which he called the earthquake norm. This norm is obtained\u0000by assigning a length to a tangent vector after such a vector is considered as\u0000an infinitesimal earthquake deformation of the surface. This induces a Finsler\u0000metric on the Teichm{\"u}ller space, called the earthquake metric. This theory\u0000was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the\u0000present paper, we study this metric from the conformal viewpoint and we adapt\u0000Thurston's theory to the case of Riemann surfaces of arbitrary genus with\u0000marked points. A complex version of the Legendre transform defined for Finsler\u0000manifolds gives an analogue of the Wolpert duality for the Weil-Petersson\u0000symplectic form, which establishes a complete analogue of Thurston's theory of\u0000the earthquake norm in the conformal setting.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.
我们提出了两个强可逆 L 空间结的例子，它们的手术从来都不是 3 球中 Khovanov 细链的双支盖。因此，这些结为 Watson 提出的强可逆 L 空间结的猜想特征提供了反例。我们还讨论了这两个结的其他特殊性质，例如，这两个 L 空间结的形式半群是实际半群。
{"title":"Two curious strongly invertible L-space knots","authors":"Kenneth L. Baker, Marc Kegel, Duncan McCoy","doi":"arxiv-2409.09833","DOIUrl":"https://doi.org/arxiv-2409.09833","url":null,"abstract":"We present two examples of strongly invertible L-space knots whose surgeries\u0000are never the double branched cover of a Khovanov thin link in the 3-sphere.\u0000Consequently, these knots provide counterexamples to a conjectural\u0000characterization of strongly invertible L-space knots due to Watson. We also\u0000discuss other exceptional properties of these two knots, for example, these two\u0000L-space knots have formal semigroups that are actual semigroups.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we explore phenomena relating to quasi-alternating surgeries on knots, where a quasi-alternating surgery on a knot is a Dehn surgery yielding the double branched cover of a quasi-alternating link. Since the double branched cover of a quasi-alternating link is an L-space, quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating surgeries except for the knots t09847 and o9_30634 which both do not have any quasi-alternating surgeries. In particular, this finishes Dunfield's classification of the L-space knots among all SnapPy census knots. In addition, we show that all asymmetric census L-space knots have exactly two quasi-alternating slopes that are consecutive integers. Similar behavior is observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and explore briefly the notion of formal L-space surgeries. This allows us to give examples of asymmetric formal L-spaces.
本文探讨了与结上的准交替手术有关的现象，其中结上的准交替手术是产生准交替链接双支盖的 Dehn 手术。由于准交替链接的双支盖是一个 L 空间，因此准交替手术是 L 空间手术的特例。我们证明，除了 t09847 和 o9_30634 这两个节点没有准交替手术之外，所有 SnapPy 普查 L 空间节点都有准交替手术。特别是，这完成了邓菲尔德对所有 SnapPy 普查结中 L 空间结的分类。此外，我们还证明了所有非对称普查 L 空间结都有两个连续整数的准交替斜率。一些贝克-吕克非对称 L 空间结也有类似行为。我们还对环状结上的准交替手术进行了分类，并简要探讨了形式 L 空间手术的概念。这使我们能够给出不对称形式 L 空间的例子。
{"title":"Quasi-alternating surgeries","authors":"Kenneth L. Baker, Marc Kegel, Duncan McCoy","doi":"arxiv-2409.09839","DOIUrl":"https://doi.org/arxiv-2409.09839","url":null,"abstract":"In this article, we explore phenomena relating to quasi-alternating surgeries\u0000on knots, where a quasi-alternating surgery on a knot is a Dehn surgery\u0000yielding the double branched cover of a quasi-alternating link. Since the\u0000double branched cover of a quasi-alternating link is an L-space,\u0000quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating\u0000surgeries except for the knots t09847 and o9_30634 which both do not have any\u0000quasi-alternating surgeries. In particular, this finishes Dunfield's\u0000classification of the L-space knots among all SnapPy census knots. In addition,\u0000we show that all asymmetric census L-space knots have exactly two\u0000quasi-alternating slopes that are consecutive integers. Similar behavior is\u0000observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and explore\u0000briefly the notion of formal L-space surgeries. This allows us to give examples\u0000of asymmetric formal L-spaces.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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