Showing papers in "Geometry & Topology in 2009"
TL;DR: In this paper, the authors developed a mathematical theory of the topological vertex of Calabi-Yau three-manifolds, a theory that was originally proposed by Aganagic, A-Klemm, M-Marino and C-Vafa.
Abstract: We have developed a mathematical theory of the topological vertex—a theory that was originally proposed by M Aganagic, A Klemm, M Marino and C Vafa on effectively computing Gromov–Witten invariants of smooth toric Calabi–Yau threefolds derived from duality between open string theory of smooth Calabi–Yau threefolds and Chern–Simons theory on three-manifolds.
179 citations
TL;DR: In this paper, a modification of Khovanov homology is proposed, which makes the theory properly functorial with respect to link cobordisms, and the authors construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of the movie moves always agree.
Abstract: We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], which makes the theory properly functorial with respect to link cobordisms. This requires introducing “disorientations” in the category of smoothings and abstract cobordisms between them used in Bar-Natan’s definition. Disorientations have “seams” separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter, Rieger and Saito’s movie moves [8; 7] always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a “local” result about tangles. Along the way, we reproduce Jacobsson’s sign table [10] for the original “unoriented theory”, with a few disagreements.
131 citations
TL;DR: In this paper, the topology of monotone Lagrangian submanifolds inside a symplectic manifold was explored by exploiting the relationships between the quantum homology of the manifold and various quantum structures associated to the Lagrangians.
Abstract: This paper explores the topology of monotone Lagrangian submanifolds L inside a symplectic manifold M by exploiting the relationships between the quantum homology of M and various quantum structures associated to the Lagrangian L.
129 citations
TL;DR: The projectivization of the Culler-Vogtmann outer space cv.FN/ provides a free group analogue of Thurston's compactification of Teichmuller space.
Abstract: h ; iW cv.FN / Curr.FN /!R 0: Here cv.FN / is the closure of unprojectivized Culler–Vogtmann Outer space cv.FN / in the equivariant Gromov–Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv.FN / consists of all very small minimal isometric actions of FN on R–trees. The projectivization of cv.FN / provides a free group analogue of Thurston’s compactification of Teichmuller space.
114 citations
TL;DR: In this article, the authors construct virtual fundamental classes for dg-manifolds whose tangent sheaves have cohomology only in degrees 0 and 1, analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi or Li and Tian [11].
Abstract: We construct virtual fundamental classes for dg‐manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi [3] or Li and Tian [11]. Our class is initially defined in K ‐theory as the class of the structure sheaf of the dg‐manifold. We compare our construction with that of [3] as well as with the original proposal of Kontsevich. We prove a Riemann‐Roch type result for dg‐ manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg‐manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg‐manifolds is the correct algebro-geometric replacement of the analytic technique of “deforming to transversal intersection”. 14F05; 14A20
113 citations
TL;DR: In this paper, a polynomial stability condition for any normal projective variety was introduced, which generalizes the Bridgeland stability condition on triangulated categories, including Simpson-stability and large volume limits.
Abstract: We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability and large volume limits of Bridgeland stability conditions. We show that the PT/DT‐correspondence relating stable pairs to Donaldson‐Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wallcrossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of one-dimensional torsion sheaves (proven recently by the same authors) is a wall-crossing formula. 14F05, 18E30; 14J32, 14D20, 14N35
104 citations
TL;DR: In this paper, the authors show that a group can be called a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an ℝ-tree.
Abstract: We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an ℝ–tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptotic cone C has countable but nontrivial fundamental group (in fact C is homeomorphic to the direct product of a tree and a circle, so π1(C)=ℤ). We show that the class of lacunary hyperbolic groups contains non–virtually cyclic elementary amenable groups, groups with all proper subgroups cyclic (Tarski monsters) and torsion groups. We show that Tarski monsters and torsion groups can have so-called graded small cancellation presentations, in which case we prove that all their asymptotic cones are hyperbolic and locally isometric to trees. This allows us to solve two problems of Druţu and Sapir and a problem of Kleiner about groups with cut points in their asymptotic cones. We also construct a finitely generated group whose divergence function is not linear but is arbitrarily close to being linear. This answers a question of Behrstock.
102 citations
TL;DR: In this article, a conjecture relating the genus-zero Gromov Witten invariants of a toric orbifold to those of a projective coarse moduli space was proposed.
Abstract: Let X be a Gorenstein orbifold with projective coarse moduli space X and let Y be a crepant resolution of X . We state a conjecture relating the genus-zero Gromov‐ Witten invariants of X to those of Y , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan‐Graber, and prove our conjecture when XD P.1;1;2/ and XD P.1;1;1;3/. As a consequence, we see that the original form of the Bryan‐Graber Conjecture holds for P.1;1;2/ but is probably false for P.1;1;1;3/. Our methods are based on mirror symmetry for toric orbifolds. 53D45; 14N35, 83E30
100 citations
TL;DR: In this article, it was shown that for each k>3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space ℝ2k which are pairwise distinct as symplectic manifolds.
Abstract: We show that for each k>3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space ℝ2k which are pairwise distinct as symplectic manifolds.
90 citations
TL;DR: In this article, the equivalence between embedded contact homology and Seiberg-Witten Floer homology was used to obtain the following improvements on the Weinstein conjecture: if Y is a closed oriented connected 3-manifold with a stable Hamiltonian structure, then R denotes the associated Reeb vector field on Y.
Abstract: We use the equivalence between embedded contact homology and Seiberg–Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3–manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y . We prove that if Y is not a T 2 –bundle over S , then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3–manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3–manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.
85 citations
TL;DR: In this paper, the Kobayashi-Hitchin correspondence between tame harmonic bundles and stable parabolic flat bundles with trivial characteristic numbers on.X;D was established and the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.
Abstract: Let X be a smooth irreducible projective complex variety with an ample line bundle L, and D be a simple normal crossing hypersurface. We establish the Kobayashi‐ Hitchin correspondence between tame harmonic bundles on X D and L ‐stable parabolic ‐flat bundles with trivial characteristic numbers on .X;D/. In particular, we obtain the quasiprojective version of the Corlette‐Simpson correspondence between flat bundles and Higgs bundles. 14J60; 53C07
TL;DR: In this article, it was shown that relatively quasiconvex subgroups of a relatively hyperbolic group have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect.
Abstract: Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case. 20F65; 20F67, 20F69
TL;DR: For a link L in a rational homology 3-sphere, the link Floer homology detects the Thurston norm of its complement as mentioned in this paper, which has been proved by Ozsvath and Szabo for links in S^3.
Abstract: We prove that, for a link L in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This result has been proved by Ozsvath and Szabo for links in S^3. As an ingredient of the proof, we show that knot Floer homology detects the genus of null-homologous links in rational homology spheres, which is a generalization of an earlier result of the author. Our argument uses the techniques due to Ozsvath and Szabo, Hedden and the author.
TL;DR: In this paper, the authors evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting and show that the conjectural equivalence with the DT vertex predicts remarkable identities.
Abstract: The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3‐folds. We evaluate the equivariant vertex for stable pairs on toric 3‐folds in terms of weighted box counting. In the toric Calabi‐Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs. 14N35; 14M25, 14D20, 14J30
TL;DR: In this article, a graphical calculus for the sl.N / link polynomial was developed and the universal sl.3/integral link homology was obtained, following an approach similar to the one adopted by Bar-Natan for the original sl.2/ integral Khovanov homology.
Abstract: In [14], Murakami, Ohtsuki and Yamada (MOY) developed a graphical calculus for the sl.N / link polynomial. In [7], Khovanov categorified the sl.3/ polynomial using singular cobordisms between webs called foams. Mackaay and Vaz [12] generalized Khovanov’s results to obtain the universal sl.3/ integral link homology, following an approach similar to the one adopted by Bar-Natan [1] for the original sl.2/ integral Khovanov homology. In [11] Khovanov and Rozansky defined a rational link homology which categorifies the sl.N / link polynomial using the theory of matrix factorizations.
TL;DR: In this article, the authors studied the geometry of near-symplectic 4-manifolds with broken Lefschetz fibrations and showed that any given broken fibration can be transformed into another which is deformation equivalent to it.
Abstract: Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4–manifolds, ie, manifolds equipped with a closed 2–form which is symplectic outside a union of embedded 1–dimensional submanifolds, and broken Lefschetz fibrations on them; see Auroux, Donaldson and Katzarkov [3] and Gay and Kirby [8]. We present a set of four moves which allow us to pass from any given broken fibration to any other which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.
TL;DR: In this article, the authors introduced ep-groupoids and polyfolds, and then generalized the Fredholm theory for M-polyfolds to the more general polyfold setting, which is the basis for our application to Symplectic Field Theory.
Abstract: This is the third in a series of papers devoted to a general Fredholm theory in a new class of spaces, called polyfolds. We first introduce ep–groupoids and polyfolds. Then we generalize the Fredholm theory, which for M–polyfolds has been presented in our paper [Geom. Funct. Anal. 18 (2009)], to the more general polyfold setting. The Fredholm theory consists of a transversality and a perturbation theory. The results form the basis for our application to Symplectic Field Theory.
TL;DR: In this article, the authors apply Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds and show that these actions are conjugate to isometric actions.
Abstract: We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3‐manifolds. Our main result is that such actions on elliptic and hyperbolic 3‐manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3‐manifolds (in the sense of Thurston) are always geometric, ie there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston [32]. 57M60, 57M50; 53C21, 53C44
TL;DR: In this paper, the authors generalize the peak reduction algorithm for free groups to a theorem about a general right-angled Artin group AΓ, and show a finite presentation for the automorphism group AutAΓ that generalizes McCool's presentation for a finite rank free group.
Abstract: We generalize the peak reduction algorithm (Whitehead’s theorem) for free groups to a theorem about a general right-angled Artin group AΓ. As an application, we find a finite presentation for the automorphism group AutAΓ that generalizes McCool’s presentation for the automorphism group of a finite rank free group. We also consider a stronger generalization of peak reduction, giving a counterexample and proving a special case.
TL;DR: In this article, it was shown that any Hamiltonian diffeomorphism of a closed rational manifold with zero first Chern class has infinitely many periodic orbits and, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension.
Abstract: The paper focuses on the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits. We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action-index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of CP n with exactly nC1 periodic orbits a stronger result holds. Namely, for such a Hamiltonian
TL;DR: In this paper, it was shown that for each n∈ℕ 0, the group ℱn∕ℱ n.5 has infinite rank, and the same result holds for the smooth concordance group.
Abstract: In 1997, T Cochran, K Orr, and P Teichner [Ann. of Math. (2) 157 (2003) 433-519] defined a filtration of the classical knot concordance group C, ⋯ ⊆ ℱ n ⊆ ⋯ ⊆ ℱ 1 ⊆ ℱ 0 . 5 ⊆ ℱ 0 ⊆ C . The filtration is important because of its strong connection to the classification of topological 4–manifolds. Here we introduce new techniques for studying C and use them to prove that, for each n∈ℕ0, the group ℱn∕ℱn.5 has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson–Gordon and Gilmer, contain slice knots.
TL;DR: In this paper, the first quasi-isometric embeddings between curve complexes are presented, which are induced by orbifold covers, and they are shown to be undistorted.
Abstract: We provide the first nontrivial examples of quasi-isometric embeddings between curve complexes; these are induced by orbifold covers. This leads to new quasi-isometric embeddings between mapping class groups. As a corollary, in the mapping class group normalizers of finite subgroups are undistorted.
TL;DR: In this paper, the authors describe other classes in H 1.M IZ/ that are represented by formal, positively weighted sums of closed integral curves of v. To set the stage for a precise statement of what is proved here, digress for a moment to introduce the set, SM, of SpinC structures on M, this a principle homogeneous space for the group H 1.
Abstract: The purpose of this article is to describe other classes in H1.M IZ/ that are represented by formal, positively weighted sums of closed integral curves of v . To set the stage for a precise statement of what is proved here, digress for a moment to introduce the set, SM , of SpinC structures on M , this a principle homogeneous space for the group H .M IZ/. Each element in S has a canonically associated Z–module, a version
TL;DR: In this article, it was shown that if S is a finite type orientable surface of negative Euler characteristic, which is not the 3-holed sphere, 4-holing sphere, or 1holed torus, then the ending lamination space of S is connected, locally path connected and cyclic.
Abstract: We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is connected, locally path connected and cyclic.
TL;DR: In this article, the authors investigated rigidity issues for polyhedral surfaces including inversive distance circle packings and showed that the discrete Laplacian operator determines a spherical polyhedral metric on a triangulated surface.
Abstract: This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson in [Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc. (2004)] as a generalization of Andreev and Thurston’s circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [Trans. Amer. Math. Soc. 363 (2011) 4757–4776] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [arXiv 0612.5714], verifying a conjecture in [arXiv 0612.5714]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.
TL;DR: In this paper, the authors construct monomorphisms between mapping class groups of surfaces, which injects the mapping class group of a closed surface into that of a different closed surface, and have quite curious behaviour.
Abstract: We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of oncepunctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.
TL;DR: In this article, it was shown that the limit set of a hyperbolic structure of bounded geometry is locally connected by constructing a natural Cannon-Thurston map, and an alternate proof and a generalization of results due to Cannon and Thurston.
Abstract: Let Nh∈H(M,P) be a hyperbolic structure of bounded geometry on a pared manifold such that each component of ∂0M=∂M−P is incompressible. We show that the limit set of Nh is locally connected by constructing a natural Cannon–Thurston map. This provides a unified treatment, an alternate proof and a generalization of results due to Cannon and Thurston, Minsky, Bowditch, Klarreich and the author.
TL;DR: In this paper, the authors present a new theory which describes the collection of all tunnels of tunnel number 1 knots in S (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures.
Abstract: We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in S (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus–2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some additional data. Up to superficial differences in definition, the final parameter of this sequence is the Scharlemann–Thompson invariant of the tunnel, and the other parameters are the Scharlemann–Thompson invariants of the intermediate tunnels produced by the constructions. We calculate the parameter sequences for tunnels of 2–bridge knots. The theory extends easily to links, and to allow equivalence of tunnels by homeomorphisms that may be orientation-reversing.
TL;DR: In this article, it was shown that if every hyperbolic group is residually finite, then every quasi-convex subgroup of a group G is separable if for every g2 GXf1g, there is some finite group F and an epimorphism W G! F so that g/O.
Abstract: We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling. 20E26, 20F67, 20F65 A group G is residually finite (or RF) if for every g2 GXf1g, there is some finite group F and an epimorphism W G! F so that . g/⁄ 1. In more sophisticated language G is RF if and only if the trivial subgroup is closed in the profinite topology on G . If H < G , then H is separable if for every g2 GXH , there is some finite group F and an epimorphism W G! F so that . g/O. H/. Equivalently, the subgroup H is separable in G if it is closed in the profinite topology on G . If every finitely generated subgroup of G is separable, G is said to be LERF or subgroup separable. If G is hyperbolic, and every quasi-convex subgroup of G is separable, we say that G is QCERF. In this paper, we show that if every hyperbolic group is RF, then every hyperbolic group is QCERF.
TL;DR: For a surface S with n marked points and fixed genus g 2, the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of S is on the order of.logn/=n as discussed by the authors.
Abstract: For a surface S with n marked points and fixed genus g 2, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of S is on the order of .logn/=n. This is in contrast with the cases of genus zero or one where the order is 1=n. 37E30; 57M99, 30F60