Mostrando recursos 1 - 20 de 227

  1. GKM-sheaves and nonorientable surface group representations

    Baird, Thomas
    Let $T$ be a compact torus and $X$ a nice compact $T$-space (say a manifold or variety). We introduce a functor assigning to $X$ a GKM-sheaf $\mathcal{F}_X$ over a GKM-hypergraph $\Gamma_X$. Under the condition that $X$ is equivariantly formal, the ring of global sections of $\mathcal{F}_X$ are identified with the equivariant cohomology, $H^*_T (X; \mathbb{C}) \cong H^0(\mathcal{F}_X)$. We show that GKM-sheaves provide a general framework able to incorporate numerous constructions in the GKM-theory literature. In the second half of the paper we apply these ideas to study the equivariant topology of the representation variety $\mathcal{R}K := \mathrm{Hom}(\pi_1 (\Sigma),K)$ under conjugation by $K$,...

  2. On cohomological obstructions for the existence of log-symplectic structures

    Mărcuţ, Ioan; Torres, Boris Osorno
    We prove that a compact log-symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of log-symplectic structures similar to those in symplectic geometry.

  3. Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere

    Hryniewicz, Umberto L.
    We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section come in families, organized as open book decompositions. As an application we obtain new global surfaces of section for the Hamiltonian dynamics on strictly convex three-dimensional energy levels.

  4. Lagrangian blow-ups, blow-downs, and applications to real packing

    Reiser, Antonio
    Given a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L$, we construct versions of the symplectic blow-up and blow-down which are defined relative to $L$. We further show that if $M$ admits an anti-symplectic involution $\phi$, i.e., a diffeomorphism such that $\phi^2 = \mathrm{Id}$ and $\phi^* \omega = - \omega$, and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an antisymplectic involution on the blow-up $\tilde{M}$ as well. We then derive a homological condition for real Lagrangian surfaces $L = \mathrm{Fix} (\phi)$ which determines when the topology of $L$ changes after a blowdown, and we use...

  5. Toric structures on bundles of projective spaces

    Fanoe, Andrew
    Recently, extending work by Karshon et al., Borisov and McDuff showed in The topology of toric symplectic manifolds that a given closed symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, in The topology of toric symplectic manifolds McDuff also showed that a product of two projective spaces $\mathbb{C}P^r \times \mathbb{C}P^s$ with any given symplectic form has a unique toric structure provided that $r, s \geq 2$. In contrast, the product $\mathbb{C}P^r \times \mathbb{C}P^1$ can be given infinitely many distinct toric structures, although only a finite number of these are compatible with each given symplectic form $\omega$. In this paper,...

  6. Infinitely many small exotic Stein fillings

    Akbulut, Selman; Yasui, Kouichi
    We show that there exist infinitely many simply connected compact Stein 4-manifolds with $b_2 = 2$ such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact $3$-manifold on their boundaries. We also describe their handlebody pictures.

  7. A remark on the Reeb flow for spheres

    Casals, Roger; Presas, Francisco
    >We prove the non-triviality of the Reeb flow for the standard contact spheres $\mathbb{S}^{2n+1}, n \neq 3$, inside the fundamental group of their contactomorphism group. The argument uses the existence of homotopically non-trivial $2$-spheres in the space of contact structures of a $3$-Sasakian manifold.

  8. Hofer Geometry and cotangent fibers

    Usher, Michael
    For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3 and compact semisimple Lie groups on the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle contains as subgroups infinitedimensional normed vector spaces that are bi-Lipschitz embedded with respect to Hofer’s metric; moreover these subgroups can be taken to consist of diffeomorphisms supported in an arbitrary neighborhood of the zero section. In fact, the orbit of a fiber of the cotangent bundle with respect to any of these subgroups is quasi-isometrically...

  9. Bypass attachments and homotopy classes of 2-plane fields in contact topology

    Huang, Yang
    We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a three-dimensional contact manifold with convex boundary, we show that the bypass triangle attachment changes the homotopy class of the contact structure relative to the boundary, and the difference is measured by the homotopy group $\pi_3(S^2)$.

  10. Smoothings of singularities and symplectic surgery

    Park, Heesang; Stipsicz, András I.
    Suppose that $C$ is a connected configuration of two-dimensional symplectic submanifolds in a symplectic 4-manifold with negative definite intersection graph $\Gamma_C$. Let $(S, 0)$ be a normal surface singularity with resolution graph $\Gamma_C$ and suppose that $W_S$ is a smoothing of $(S, 0)$. We show that if we replace an appropriate neighborhood of $C$ with $W_S$, then the resulting 4-manifold admits a symplectic structure. The operation generalizes the rational blow-down operation of Fintushel-Stern, and therefore our result extends Symington's theorem about symplectic rational blow-downs.

  11. Bilinearized Legendrian contact homology and the augmentation category

    Bourgeois, Frédéric; Chantraine, Baptiste
    In this paper, we construct an $\mathcal{A}_{\infty}$-category associated to a Legendrian submanifold of a jet space. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(\Lambda)$ and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearized Legendrian contact homology. Those are constructed as a generalization of linearized Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order composition maps in the category and generalizes the idea of G. Civan, P. Koprowski, J. Etnyre, J.M. Sabloff and A. Walker, Product structures for Legendrian contact...

  12. Symplectic homology of disc cotangent bundles of domains in Euclidean space

    Irie, Kei
    Let $V$ be a bounded domain with smooth boundary in $\mathbb{R}^n$, and $D^*V$ denote its disc cotangent bundle. We compute symplectic homology of $D^*V$, in terms of relative homology of loop spaces on the closure of $V$. We use this result to show that the Floer-Hofer-Wysocki capacity of $D^*V$ is between $2r(V)$ and $2(n + 1)r(V)$, where $r(V)$ denotes the inradius of $V$. As an application, we study periodic billiard trajectories on $V$.

  13. Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

    Hamilton, Mark D.; Konno, Hiroshi
    In this paper, we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand–Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr–Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.

  14. Generalized reduction and pure spinors

    Drummond, T.
    We study reduction of Dirac structures as developed by H. Bursztyn et al. from the point of view of pure spinors. We describe explicitly the pure spinor line bundle of the reduced Dirac structure. We also obtain results on reduction of generalized Calabi-Yau structures.

  15. Square roots of Hamiltonian diffeomorphisms

    Albers, Peter; Frauenfelder, Urs
    In this paper, we prove that on any closed symplectic manifold there exists an arbitrarily $C^\infty$-small Hamiltonian diffeomorphism not admitting a square root.

  16. Open books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristic

    Chiang, River; Ding, Fan; van Koert, Otto
    We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fibered Dehn twists are not symplectically isotopic to the identity relative to the boundary. This complements results of Biran and Giroux.

  17. The closure of the symplectic cone of elliptic surfaces

    Hamilton, M. J. D.
    The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kähler surface. We consider the case of the elliptic surfaces $E(n)$ and focus on a slightly weaker conjecture for the closure of the symplectic cone. We prove this conjecture in the case of the spin surfaces $E(2m)$ using inflation and the action of self-diffeomorphisms of the elliptic surface. An additional obstruction appears in the non-spin case.

  18. On the anti-diagonal filtration for the Heegard Floer chain complex of a branched double-cover

    Tweedy, Eamonn
    Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group $Kh_{\rm symp,~inv}(K)$ for a knot $K \subset S^{3}$, as well as a spectral sequence converging to the Heegaard Floer homology group $\widehat{HF}(\Sigma (K) \# (S^2 \times S^1))$ with $E^1$-page isomorphic to a factor of $Kh_{\rm symp,~inv}(K)$. There the authors proved that $Kh_{\rm symp,~inv}$ is a knot invariant. We show here that the higher pages of their spectral sequence are knot invariants also.

  19. Removal of singularities and Gromov compactness for symplectic vortices

    Ott, Andreas
    We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.

  20. A bordered Legendrian contact algebra

    Harper, John G.; Sullivan, Michael G.
    In A bordered Chekanov–Eliashberg algebra, Sivek proves a "van Kampen" decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard contact $\mathbb{R}^3$. We prove an analogous result for the holomorphic curve version of the Legendrian contact algebra of certain Legendrians submanifolds in standard contact $J^1(M)$. This includes all one- and two-dimensional Legendrians, and some higher-dimensional ones. We present various applications including a Mayer-Vietoris sequence for linearized contact homology similar to A bordered Chekanov–Eliashberg algebra and a connect sum formula for the augmentation variety introduced in L. Ng, Framed knot contact homology. The main tool is...

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