Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
{"title":"Unknotting with a single twist","authors":"S. Allen, C. Livingston","doi":"10.4171/LEM/66-3/4-10","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-10","url":null,"abstract":"Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124661179","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}
For any finite group $G$, Thompson proved that, for each $chiin {rm Irr}(G)$, $chi(g)$ is a root of unity or zero for more than a third of the elements $gin G$, and Gallagher proved that, for each larger than average class $g^G$, $chi(g)$ is a root of unity or zero for more than a third of the irreducible characters $chiin {rm Irr}(G)$. We show that in many cases"more than a third"can be replaced by"more than half".
{"title":"Zeros and roots of unity in character tables","authors":"A. Miller","doi":"10.4171/lem/1042","DOIUrl":"https://doi.org/10.4171/lem/1042","url":null,"abstract":"For any finite group $G$, Thompson proved that, for each $chiin {rm Irr}(G)$, $chi(g)$ is a root of unity or zero for more than a third of the elements $gin G$, and Gallagher proved that, for each larger than average class $g^G$, $chi(g)$ is a root of unity or zero for more than a third of the irreducible characters $chiin {rm Irr}(G)$. We show that in many cases\"more than a third\"can be replaced by\"more than half\".","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":" 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141220158","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 gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some of it is apparently not published at all.
{"title":"Collared and non-collared manifold boundaries","authors":"M. Baillif","doi":"10.4171/LEM/1024","DOIUrl":"https://doi.org/10.4171/LEM/1024","url":null,"abstract":"We gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some of it is apparently not published at all.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126524012","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}
For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $widetilde M$ is a $Gamma$-regular covering space of $M$. Let $langle alpha rangle$ be the conjugacy class of a non-identity element $alphain Gamma$. Suppose ${Gamma_i}$ is a sequence of finite-index normal subgroups of $Gamma$ that distinguishes $langle alpha rangle$. Let $pi_{Gamma_i}$ be the quotient map from $Gamma$ to $Gamma/Gamma_i$ and $langle pi_{Gamma_i}(alpha) rangle$ the conjugacy class of $pi_{Gamma_i}(alpha)$ in $Gamma/Gamma_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $widetilde M$ at the conjugacy class $langle alpha rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{Gamma_i}$ at the conjugacy class $langle pi_{Gamma_i}(alpha) rangle$, where $M_{Gamma_i}= widetilde M/Gamma_i$ is the finite-sheeted covering space of $M$ determined by $Gamma_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{Gamma_i}$ at the conjugacy class $langle pi_{Gamma_i}(alpha) rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $Gamma$.
{"title":"Approximations of delocalized eta invariants by their finite analogues","authors":"Jinmin Wang, Zhizhang Xie, Guoliang Yu","doi":"10.4171/lem/1048","DOIUrl":"https://doi.org/10.4171/lem/1048","url":null,"abstract":"For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $widetilde M$ is a $Gamma$-regular covering space of $M$. Let $langle alpha rangle$ be the conjugacy class of a non-identity element $alphain Gamma$. Suppose ${Gamma_i}$ is a sequence of finite-index normal subgroups of $Gamma$ that distinguishes $langle alpha rangle$. Let $pi_{Gamma_i}$ be the quotient map from $Gamma$ to $Gamma/Gamma_i$ and $langle pi_{Gamma_i}(alpha) rangle$ the conjugacy class of $pi_{Gamma_i}(alpha)$ in $Gamma/Gamma_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $widetilde M$ at the conjugacy class $langle alpha rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{Gamma_i}$ at the conjugacy class $langle pi_{Gamma_i}(alpha) rangle$, where $M_{Gamma_i}= widetilde M/Gamma_i$ is the finite-sheeted covering space of $M$ determined by $Gamma_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{Gamma_i}$ at the conjugacy class $langle pi_{Gamma_i}(alpha) rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $Gamma$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"260 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132327349","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 paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine-Stieltjes theory of linear differential equations with polynomial coefficients.
{"title":"On existence of quasi-Strebel structures for meromorphic $k$-differentials","authors":"B. Shapiro, Guillaume Tahar","doi":"10.4171/lem/1008","DOIUrl":"https://doi.org/10.4171/lem/1008","url":null,"abstract":"In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine-Stieltjes theory of linear differential equations with polynomial coefficients.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"66 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":"126084287","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 Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomial values and prime numbers. In the same vein we investigate the common divisors of values $P_1(n),ldots, P_s(n)$ of several polynomials. We deduce this coprime version of the Schinzel Hypothesis: under some natural assumption, coprime polynomials assume coprime values at infinitely many integers. Consequences include a version "modulo an integer" of the original Schinzel Hypothesis, with the Goldbach conjecture, again modulo an integer, as a special case.
{"title":"Prime and coprime values of polynomials","authors":"Arnaud Bodin, P. Dèbes, S. Najib","doi":"10.4171/lem/66-1/2-9","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-9","url":null,"abstract":"The Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomial values and prime numbers. In the same vein we investigate the common divisors of values $P_1(n),ldots, P_s(n)$ of several polynomials. We deduce this coprime version of the Schinzel Hypothesis: under some natural assumption, coprime polynomials assume coprime values at infinitely many integers. Consequences include a version \"modulo an integer\" of the original Schinzel Hypothesis, with the Goldbach conjecture, again modulo an integer, as a special case.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"60 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":"128243611","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 [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $mathrm{Homeo}(D,partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $mathrm{Homeo}(D,partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.
{"title":"On James Hyde’s example of non-orderable subgroup of $mathrm{Homeo}(D,partial D)$","authors":"Michele Triestino","doi":"10.4171/LEM/66-3/4-5","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-5","url":null,"abstract":"In [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $mathrm{Homeo}(D,partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $mathrm{Homeo}(D,partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123276488","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 show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable ordinal, such as amenability and Roelcke-precompactness.
{"title":"The homeomorphism group of the first uncountable ordinal","authors":"Maxime Gheysens","doi":"10.4171/lem/1006","DOIUrl":"https://doi.org/10.4171/lem/1006","url":null,"abstract":"We show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable ordinal, such as amenability and Roelcke-precompactness.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125904948","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 $M = Gamma backslash text{SL}(2,mathbb{R})$ be a compact quotient of $text{SL}(2,mathbb{R})$ equipped with the normalized Haar measure $text{vol}$, and let ${h_t}_{t in mathbb{R}}$ denote the horocycle flow on $M$. Given $p in M$ and $W in mathfrak{sl}_2(mathbb{R}) setminus {0}$ not parallel to the generator of the horocycle flow, let $gamma_{p}^W$ denote the probability measure uniformly distributed along the arc $s mapsto p exp(sW)$ for $0leq s leq 1$. We establish quantitative estimates for the rate of convergence of $[(h_t)_{ast} gamma_{p}^W](f)$ to $text{vol}(f)$ for sufficiently smooth functions $f$. Our result is based on the work of Bufetov and Forni [2], together with a crucial geometric observation. As a corollary, we provide an alternative proof of Ratner's theorem on quantitative mixing for the horocycle flow.
设$M = Gamma backslash text{SL}(2,mathbb{R})$为具有归一化哈尔测度$text{vol}$的$text{SL}(2,mathbb{R})$的紧商，设${h_t}_{t in mathbb{R}}$为$M$上的环流。假定$p in M$和$W in mathfrak{sl}_2(mathbb{R}) setminus {0}$不平行于环形流的发生器，设$gamma_{p}^W$表示$0leq s leq 1$沿弧线$s mapsto p exp(sW)$均匀分布的概率测度。对于足够光滑的函数$f$，我们建立了$[(h_t)_{ast} gamma_{p}^W](f)$到$text{vol}(f)$收敛速率的定量估计。我们的结果是基于Bufetov和Forni[2]的工作，以及一个关键的几何观测。作为一个推论，我们提供了关于环形流定量混合的拉特纳定理的另一种证明。
{"title":"Quantitative equidistribution of horocycle push-forwards of transverse arcs","authors":"Davide Ravotti","doi":"10.4171/lem/66-1/2-7","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-7","url":null,"abstract":"Let $M = Gamma backslash text{SL}(2,mathbb{R})$ be a compact quotient of $text{SL}(2,mathbb{R})$ equipped with the normalized Haar measure $text{vol}$, and let ${h_t}_{t in mathbb{R}}$ denote the horocycle flow on $M$. Given $p in M$ and $W in mathfrak{sl}_2(mathbb{R}) setminus {0}$ not parallel to the generator of the horocycle flow, let $gamma_{p}^W$ denote the probability measure uniformly distributed along the arc $s mapsto p exp(sW)$ for $0leq s leq 1$. We establish quantitative estimates for the rate of convergence of $[(h_t)_{ast} gamma_{p}^W](f)$ to $text{vol}(f)$ for sufficiently smooth functions $f$. Our result is based on the work of Bufetov and Forni [2], together with a crucial geometric observation. As a corollary, we provide an alternative proof of Ratner's theorem on quantitative mixing for the horocycle flow.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134442387","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}
Serge Cantat, D. Cerveau, Vincent Guirardel, J. Souto
We construct embeddings of surface groups into the group of germs of analytic diffeomorphisms in one variable.
我们将曲面群的嵌入构造为单变量解析微分同态的胚芽群。
{"title":"Surface groups in the group of germs of analytic diffeomorphisms in one variable","authors":"Serge Cantat, D. Cerveau, Vincent Guirardel, J. Souto","doi":"10.4171/lem/66-1/2-6","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-6","url":null,"abstract":"We construct embeddings of surface groups into the group of germs of analytic diffeomorphisms in one variable.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115525239","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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