Gil Bor, L. Hernández-Lamoneda, Valent'in Jim'enez-Desantiago, Luis Montejano-Peimbert
Let $V$ be a Banach space where for fixed $n$, $1
设$V$是一个巴拿赫空间，对于固定的$n$, $1
{"title":"On the isometric conjecture of Banach","authors":"Gil Bor, L. Hernández-Lamoneda, Valent'in Jim'enez-Desantiago, Luis Montejano-Peimbert","doi":"10.2140/gt.2021.25.2621","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2621","url":null,"abstract":"Let $V$ be a Banach space where for fixed $n$, $1<n<dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ and all $V$. In this paper we give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133.$ Our proof relies on a new characterization of ellipsoids in ${mathbb{R}}^n$, $ngeq 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"19 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91352097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A subgroup $Hleq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a finite valence bushy tree. Using work of Mosher, we generalise a result of Farb-Mosher to show that for many surface group extensions $Gamma_L$, any group quasi-isometric to $Gamma_L$ is virtually isomorphic to $Gamma_L$. We also prove quasi-isometric rigidity for the class of finitely presented $mathbb{Z}$-by-($infty$ ended) groups.
{"title":"The geometry of groups containing almost normal subgroups","authors":"Alexander Margolis","doi":"10.2140/gt.2021.25.2405","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2405","url":null,"abstract":"A subgroup $Hleq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a finite valence bushy tree. Using work of Mosher, we generalise a result of Farb-Mosher to show that for many surface group extensions $Gamma_L$, any group quasi-isometric to $Gamma_L$ is virtually isomorphic to $Gamma_L$. We also prove quasi-isometric rigidity for the class of finitely presented $mathbb{Z}$-by-($infty$ ended) groups.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"52 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91393612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the $frac12$-snowflake of a finite-dimensional normed space $(X,|cdot|_X)$ embeds into a Hilbert space with quadratic average distortion $$OBig(sqrt{log mathrm{dim}(X)}Big).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $Dgeqslant 1$ into $(X,|cdot|_X)$, then necessarily $mathrm{dim}(X)geqslant n^{Omega(1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $mathrm{dim}(X)gtrsim (log n)^2/D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matouv{s}ek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodr{'{i}}guez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).
{"title":"An average John theorem","authors":"A. Naor","doi":"10.2140/gt.2021.25.1631","DOIUrl":"https://doi.org/10.2140/gt.2021.25.1631","url":null,"abstract":"We prove that the $frac12$-snowflake of a finite-dimensional normed space $(X,|cdot|_X)$ embeds into a Hilbert space with quadratic average distortion $$OBig(sqrt{log mathrm{dim}(X)}Big).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $Dgeqslant 1$ into $(X,|cdot|_X)$, then necessarily $mathrm{dim}(X)geqslant n^{Omega(1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $mathrm{dim}(X)gtrsim (log n)^2/D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matouv{s}ek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodr{'{i}}guez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"28 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81444667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $mathbb{C}mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $tilde{M}$ that is a non-trivial bundle over a compact surface.
{"title":"Compact hyperbolic manifolds without spin structures","authors":"B. Martelli, Stefano Riolo, Leone Slavich","doi":"10.2140/gt.2020.24.2647","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2647","url":null,"abstract":"We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $mathbb{C}mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $tilde{M}$ that is a non-trivial bundle over a compact surface.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"21 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78864446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We import into homotopy theory the algebro-geometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava $K$-theory of height $d$, we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of $K(mathbb Z_p, d+1)$. Coupling these ideas to work of Westerland, we give a "Snaith's theorem" for the Iwasawa extension of the $K(d)$-local sphere.
{"title":"Coalgebraic formal curve spectra and spectral jet spaces","authors":"E. Peterson","doi":"10.2140/gt.2020.24.1","DOIUrl":"https://doi.org/10.2140/gt.2020.24.1","url":null,"abstract":"We import into homotopy theory the algebro-geometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava $K$-theory of height $d$, we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of $K(mathbb Z_p, d+1)$. Coupling these ideas to work of Westerland, we give a \"Snaith's theorem\" for the Iwasawa extension of the $K(d)$-local sphere.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"3 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82395175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these three properties for more than 300,000 hyperbolic rational homology 3-spheres. New or much improved methods for studying each of these properties form the bulk of the paper, including a new combinatorial criterion, called a foliar orientation, for showing that a 3-manifold has a taut foliation.
{"title":"Floer homology, group orderability, and taut\u0000foliations of hyperbolic 3–manifolds","authors":"N. Dunfield","doi":"10.2140/gt.2020.24.2075","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2075","url":null,"abstract":"This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these three properties for more than 300,000 hyperbolic rational homology 3-spheres. New or much improved methods for studying each of these properties form the bulk of the paper, including a new combinatorial criterion, called a foliar orientation, for showing that a 3-manifold has a taut foliation.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"12 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76179233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3 surfaces and surfaces of general type with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants respectively. The dimension 1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers which is of independent interest.
{"title":"Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics","authors":"D. Oprea, R. Pandharipande","doi":"10.2140/gt.2021.25.3425","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3425","url":null,"abstract":"We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3 surfaces and surfaces of general type with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants respectively. The dimension 1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers which is of independent interest.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"65 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83319995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Bonsante, J. Danciger, Sara Maloni, Jean-Marc Schlenker
Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature $K in [-1,0)$ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of $K$, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes.
{"title":"The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometry","authors":"F. Bonsante, J. Danciger, Sara Maloni, Jean-Marc Schlenker","doi":"10.2140/gt.2021.25.2827","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2827","url":null,"abstract":"Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature $K in [-1,0)$ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of $K$, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"88 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86962064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Irving Dai, Jennifer Hom, Matthew Stoffregen, L. Truong
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.
{"title":"More concordance homomorphisms from knot Floer homology","authors":"Irving Dai, Jennifer Hom, Matthew Stoffregen, L. Truong","doi":"10.2140/gt.2021.25.275","DOIUrl":"https://doi.org/10.2140/gt.2021.25.275","url":null,"abstract":"We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"184 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73001315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that coinvariants of modules over conformal vertex algebras give rise to quasi-coherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by Tsuchiya-Kanie, Tsuchiya-Ueno-Yamada, and extend work of a number of researchers. The sheaves carry a twisted logarithmic D-module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.
{"title":"Conformal blocks from vertex algebras and their\u0000connections on ℳg,n","authors":"Chiara Damiolini, A. Gibney, Nicola Tarasca","doi":"10.2140/gt.2021.25.2235","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2235","url":null,"abstract":"We show that coinvariants of modules over conformal vertex algebras give rise to quasi-coherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by Tsuchiya-Kanie, Tsuchiya-Ueno-Yamada, and extend work of a number of researchers. The sheaves carry a twisted logarithmic D-module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"93 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84157699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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