A problem from thermodynamic formalism for countable symbolic Markov chains is considered. It concerns asymptotic behavior of the equilibrium measures corresponding to increasing sequences of finite submatrices of an infinite nonnegative matrix A A when these sequences converge to A A . After reviewing the results obtained up to now, a solution of the problem is given for a new matrix class. The geometric language of loaded graphs is used, instead of the matrix language.
{"title":"Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: New examples","authors":"B. Gurevich","doi":"10.1090/conm/772/15487","DOIUrl":"https://doi.org/10.1090/conm/772/15487","url":null,"abstract":"A problem from thermodynamic formalism for countable symbolic Markov chains is considered. It concerns asymptotic behavior of the equilibrium measures corresponding to increasing sequences of finite submatrices of an infinite nonnegative matrix \u0000\u0000 \u0000 A\u0000 A\u0000 \u0000\u0000 when these sequences converge to \u0000\u0000 \u0000 A\u0000 A\u0000 \u0000\u0000. After reviewing the results obtained up to now, a solution of the problem is given for a new matrix class. The geometric language of loaded graphs is used, instead of the matrix language.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"72 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127778014","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 construct a link in the 33-space that is not isotopic to any PL link (non-ambiently). In fact, we show that there exist uncountably many II-equivalence classes of links.
The paper also includes some observations on Cochran’s invariants βibeta _i.
我们在33 -空间中构造了一个不与任何PL链路(非环境)同位素的链路。事实上，我们证明了存在不可数的I - I -等价类。本文还包括对Cochran不变量β i β _i的一些观察。
{"title":"Topological isotopy and Cochran’s derived invariants","authors":"S. A. Melikhov","doi":"10.1090/conm/772/15493","DOIUrl":"https://doi.org/10.1090/conm/772/15493","url":null,"abstract":"<p>We construct a link in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\u0000 <mml:semantics>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-space that is not isotopic to any PL link (non-ambiently). In fact, we show that there exist uncountably many <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivalence classes of links.</p>\u0000\u0000<p>The paper also includes some observations on Cochran’s invariants <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta Subscript i\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>β<!-- β --></mml:mi>\u0000 <mml:mi>i</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">beta _i</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116440782","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 original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.
{"title":"Amenability of groupoids and asymptotic invariance of convolution powers","authors":"Theo Buhler, V. Kaimanovich","doi":"10.1090/conm/772/15482","DOIUrl":"https://doi.org/10.1090/conm/772/15482","url":null,"abstract":"The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"114 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120977027","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}
This is an expanded version of the talk given by the first author at the conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose of this talk was to explain our current results on the classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. A detailed exposition will appear elsewhere.
{"title":"Anti-symplectic involutions on rational symplectic 4-manifolds","authors":"V. Kharlamov, V. Shevchishin","doi":"10.1090/conm/772/15488","DOIUrl":"https://doi.org/10.1090/conm/772/15488","url":null,"abstract":"This is an expanded version of the talk given by the first author at the conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose of this talk was to explain our current results on the classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. A detailed exposition will appear elsewhere.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114862301","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 2014 A. Degtyarev, I. Itenberg, and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type I (over a base B B of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for B = P 1 B=mathbb {P}^1 obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type II over B = P 1 B=mathbb {P}^1 .
在2014年A。Degtyarev, I. Itenberg和作者用足够简单图的组合学描述了I型(任意属B B上)的最大弯曲实三角曲线的纤维等变变形，并对B= P 1 B=mathbb {P}^1给出了这类曲线的完全分类。本文将上述结果推广到B= P 1 B=mathbb {P}^1上II型最大弯曲实三角曲线。
{"title":"Maximally inflected real trigonal curves on Hirzebruch surfaces","authors":"V. Zvonilov","doi":"10.1090/conm/772/15498","DOIUrl":"https://doi.org/10.1090/conm/772/15498","url":null,"abstract":"In 2014 A. Degtyarev, I. Itenberg, and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type I (over a base \u0000\u0000 \u0000 B\u0000 B\u0000 \u0000\u0000 of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for \u0000\u0000 \u0000 \u0000 B\u0000 =\u0000 \u0000 \u0000 P\u0000 \u0000 1\u0000 \u0000 \u0000 B=mathbb {P}^1\u0000 \u0000\u0000 obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type II over \u0000\u0000 \u0000 \u0000 B\u0000 =\u0000 \u0000 \u0000 P\u0000 \u0000 1\u0000 \u0000 \u0000 B=mathbb {P}^1\u0000 \u0000\u0000.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"159 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123169040","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 survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley–Reisner rings of simplicial complexes.
{"title":"Higher order Massey products and applications","authors":"I. Limonchenko, D. Millionshchikov","doi":"10.1090/conm/772/15491","DOIUrl":"https://doi.org/10.1090/conm/772/15491","url":null,"abstract":"In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley–Reisner rings of simplicial complexes.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116555657","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}
Eunjeong Lee, M. Masuda, Seonjeong Park, Jongbaek Song
The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
{"title":"Poincaré polynomials of generic torus orbit closures in Schubert varieties","authors":"Eunjeong Lee, M. Masuda, Seonjeong Park, Jongbaek Song","doi":"10.1090/conm/772/15490","DOIUrl":"https://doi.org/10.1090/conm/772/15490","url":null,"abstract":"The closure of a generic torus orbit in the flag variety \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 /\u0000 \u0000 B\u0000 \u0000 G/B\u0000 \u0000\u0000 of type \u0000\u0000 \u0000 A\u0000 A\u0000 \u0000\u0000 is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 /\u0000 \u0000 B\u0000 \u0000 G/B\u0000 \u0000\u0000. When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123404073","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 a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima–Yamasaki η eta -function (in the univariate case) and Cochran invariants, on the other hand.
给出了积分同调球上彩色连杆的一个新的不变量斜率，并利用该不变量补全了两连杆拼接的签名公式。我们开发了许多计算斜率的方法和一些消失的结果。此外，我们讨论了斜率的一致性不变性，并建立了它与Conway多项式、Kojima-Yamasaki η eta -函数(在单变量情况下)和Cochran不变量的密切关系。
{"title":"Slopes of links and signature formulas","authors":"A. Degtyarev, V. Florens, Ana G. Lecuona","doi":"10.1090/conm/772/15483","DOIUrl":"https://doi.org/10.1090/conm/772/15483","url":null,"abstract":"We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima–Yamasaki \u0000\u0000 \u0000 η\u0000 eta\u0000 \u0000\u0000-function (in the univariate case) and Cochran invariants, on the other hand.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116902998","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 suggest a short proof of O.Benoist and O.Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface $X$ of dimension $ge 2$ the real lines on $X$ generate the whole group $H_1(X(Bbb R);Bbb Z/2)$.
本文给出了o.b inoist定理和O.Wittenberg定理(arXiv:1907.10859)的一个简短证明，证明了对于每一个维数为$ ge 2的实非奇异三次超曲面$X$， $X$上的实直线生成整个群$H_1(X(Bbb R);Bbb Z/2)$。
{"title":"The first homology of a real cubic is generated by lines","authors":"S. Finashin, V. Kharlamov","doi":"10.1090/conm/772/15485","DOIUrl":"https://doi.org/10.1090/conm/772/15485","url":null,"abstract":"We suggest a short proof of O.Benoist and O.Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface $X$ of dimension $ge 2$ the real lines on $X$ generate the whole group $H_1(X(Bbb R);Bbb Z/2)$.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122692186","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 describe the basic Dolbeault cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a DGA model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kähler (equivalently, polytopal) case using a foliated analogue of toric blow-up.
{"title":"Dolbeault cohomology of complex manifolds with torus action","authors":"Roman Krutowski, T. Panov","doi":"10.1090/conm/772/15489","DOIUrl":"https://doi.org/10.1090/conm/772/15489","url":null,"abstract":"We describe the basic Dolbeault cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a DGA model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kähler (equivalently, polytopal) case using a foliated analogue of toric blow-up.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131341816","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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