Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,mu_i geq-epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $epsilon_ito 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on. Instead, we introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,mu_i geq -epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$.
{"title":"dp–convergence and 𝜖–regularity theorems for\u0000entropy and scalar curvature lower bounds","authors":"Man-Chun Lee, A. Naber, Robin Neumayer","doi":"10.2140/gt.2023.27.227","DOIUrl":"https://doi.org/10.2140/gt.2023.27.227","url":null,"abstract":"Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,mu_i geq-epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $epsilon_ito 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on. Instead, we introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,mu_i geq -epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115276321","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 article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^sharp,T^ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T^sharp,T^ast$ are of type $F_infty$ and that $mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.
{"title":"Asymptotically rigid mapping class\u0000groups, I : Finiteness properties of braided Thompson’s and Houghton’s\u0000groups","authors":"A. Genevois, Anne Lonjou, Christian Urech","doi":"10.2140/gt.2022.26.1385","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1385","url":null,"abstract":"This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^sharp,T^ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T^sharp,T^ast$ are of type $F_infty$ and that $mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132040476","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 that the integral cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of $1$-cycles. Equivalently, it is of strong coniveau 1 in the sense of Benoist-Ottem.
{"title":"On the coniveau of rationally connected threefolds","authors":"C. Voisin","doi":"10.2140/gt.2022.26.2731","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2731","url":null,"abstract":"We prove that the integral cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of $1$-cycles. Equivalently, it is of strong coniveau 1 in the sense of Benoist-Ottem.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125494636","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}
Antonio Alfieri, John A. Baldwin, Irving Dai, Steven Sivek
We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $smash{I^#(Y)}$ is isomorphic to the Heegaard Floer homology $smash{widehat{mathit{HF}}(Y; mathbb{C})}$. This class of 3-manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres$unicode{x2014}$with base $mathbb{RP}^2$$unicode{x2014}$directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.
{"title":"Instanton Floer homology of almost-rational plumbings","authors":"Antonio Alfieri, John A. Baldwin, Irving Dai, Steven Sivek","doi":"10.2140/gt.2022.26.2237","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2237","url":null,"abstract":"We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $smash{I^#(Y)}$ is isomorphic to the Heegaard Floer homology $smash{widehat{mathit{HF}}(Y; mathbb{C})}$. This class of 3-manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres$unicode{x2014}$with base $mathbb{RP}^2$$unicode{x2014}$directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116979870","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 map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parameterized generalization of Fei Han's realization of the Chern character in K-theory as dimensional reduction for $1|1$-dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2|1$-dimensional tori and the derived geometry of complex analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $mathcal{N}=(0,1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.
{"title":"Chern characters for supersymmetric field theories","authors":"Daniel Berwick-Evans","doi":"10.2140/gt.2023.27.1947","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1947","url":null,"abstract":"We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parameterized generalization of Fei Han's realization of the Chern character in K-theory as dimensional reduction for $1|1$-dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2|1$-dimensional tori and the derived geometry of complex analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $mathcal{N}=(0,1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"193 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115212618","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 relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -- including mapping class groups of surfaces -- are coarsely injective and coarsely injective groups are strongly shortcut. Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, have finitely many conjugacy classes of finite subgroups, and that their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.
{"title":"Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity","authors":"T. Haettel, Nima Hoda, H. Petyt","doi":"10.2140/gt.2023.27.1587","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1587","url":null,"abstract":"We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -- including mapping class groups of surfaces -- are coarsely injective and coarsely injective groups are strongly shortcut. \u0000Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, have finitely many conjugacy classes of finite subgroups, and that their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126005560","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 study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we prove that certain pairs of topological $4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries, and equivalent equivariant intersection forms, are homeomorphic.
{"title":"Embedded surfaces with infinite cyclic knot group","authors":"Anthony Conway, Mark Powell","doi":"10.2140/gt.2023.27.739","DOIUrl":"https://doi.org/10.2140/gt.2023.27.739","url":null,"abstract":"We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we prove that certain pairs of topological $4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries, and equivalent equivariant intersection forms, are homeomorphic.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132623886","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 that the hierarchical hulls of finite sets of points in mapping class groups and Teichmuller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmuller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of "colorable" hierarchically hyperbolic spaces and groups.
{"title":"Stable cubulations, bicombings, and barycenters","authors":"Matthew G. Durham, Y. Minsky, A. Sisto","doi":"10.2140/gt.2023.27.2383","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2383","url":null,"abstract":"We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmuller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmuller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of \"colorable\" hierarchically hyperbolic spaces and groups.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115267816","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 define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of $mu_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space $mathbb{CP}^{infty}_{mu_p}$, which we prove to be homologically even in a sense generalizing recent $C_2$-equivariant work on conjugation spaces. We prove that the height $p-1$ Morava $E$-theory is $mu_p$-oriented and that $mathrm{tmf}(2)$ is $mu_3$-oriented. We explain how a single equivariant map $v_1^{mu_p}:S^{2rho} to Sigma^{infty} mathbb{CP}^{infty}_{mu_p}$ completely generates the homotopy of $E_{p-1}$ and $mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.
{"title":"Odd primary analogs of real orientations","authors":"Jeremy Hahn, Andrew Senger, D. Wilson","doi":"10.2140/gt.2023.27.87","DOIUrl":"https://doi.org/10.2140/gt.2023.27.87","url":null,"abstract":"We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of $mu_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space $mathbb{CP}^{infty}_{mu_p}$, which we prove to be homologically even in a sense generalizing recent $C_2$-equivariant work on conjugation spaces. \u0000We prove that the height $p-1$ Morava $E$-theory is $mu_p$-oriented and that $mathrm{tmf}(2)$ is $mu_3$-oriented. We explain how a single equivariant map $v_1^{mu_p}:S^{2rho} to Sigma^{infty} mathbb{CP}^{infty}_{mu_p}$ completely generates the homotopy of $E_{p-1}$ and $mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133822228","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 BSTRACT . The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan’s theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin-Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.
{"title":"Discrete conformal geometry of polyhedral surfaces and its convergence","authors":"F. Luo, Jian Sun, Tianqi Wu","doi":"10.2140/gt.2022.26.937","DOIUrl":"https://doi.org/10.2140/gt.2022.26.937","url":null,"abstract":"A BSTRACT . The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan’s theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin-Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"342 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134144928","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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