Mostrando recursos 1 - 20 de 1.311

  1. Smooth factors of projective actions of higher-rank lattices and rigidity

    Gorodnik, Alexander; Spatzier, Ralf
    We study smooth factors of the standard actions of lattices in higher-rank semisimple Lie groups on flag manifolds. Under a mild condition on the existence of a single differentiable sink, we show that these factors are [math] –conjugate to the standard actions on flag manifolds.

  2. Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces

    Asok, Aravind; Hoyois, Marc; Wendt, Matthias
    We establish a relative version of the abstract “affine representability” theorem in [math] –homotopy theory from part I of this paper. We then prove some [math] –invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass–Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in [math] –homotopy theory.

  3. Markov numbers and Lagrangian cell complexes in the complex projective plane

    Evans, Jonathan David; Smith, Ivan
    We study Lagrangian embeddings of a class of two-dimensional cell complexes [math] into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type [math] (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into [math] then [math] is a Markov number and we completely characterise [math] . We also show that a collection of Lagrangian pinwheels [math] , [math] , cannot be made disjoint unless [math] and the [math] form part of a Markov triple. These results are the symplectic analogue of a...

  4. Topology of closed hypersurfaces of small entropy

    Bernstein, Jacob; Wang, Lu
    We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in [math] with entropy less than or equal to that of [math] , the round cylinder in [math] , are diffeomorphic to [math] .

  5. Pixton's double ramification cycle relations

    Clader, Emily; Janda, Felix
    We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on [math] vanishes in codimension beyond [math] . This yields a collection of tautological relations in the Chow ring of [math] . We describe, furthermore, how these relations can be obtained from Pixton’s [math] –spin relations via localization on the moduli space of stable maps to an orbifold projective line.

  6. Long-time behavior of $3$–dimensional Ricci flow, D: Proof of the main results

    Bamler, Richard H
    This is the fourth and last part of a series of papers on the long-time behavior of [math] –dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes nonsingular eventually and the curvature is bounded by [math] . The second result provides a qualitative description of the geometry as [math] .

  7. Long-time behavior of $3$–dimensional Ricci flow, C: $3$–manifold topology and combinatorics of simplicial complexes in $3$–manifolds

    Bamler, Richard H
    In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of [math] –manifolds. The main part is devoted to the construction of certain simplicial complexes in a given [math] –manifold that exhibit useful intersection properties with embedded, incompressible solid tori. ¶ This paper is purely topological in nature and Ricci flows will not be used.

  8. Long-time behavior of $3$–dimensional Ricci flow, B: Evolution of the minimal area of simplicial complexes under Ricci flow

    Bamler, Richard H
    In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a [math] –manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. ¶ We remark that in this paper we will mostly be dealing with nonsingular Ricci flows. The existence of surgeries will not play an important role.

  9. Long-time behavior of $3$–dimensional Ricci flow, A: Generalizations of Perelman's long-time estimates

    Bamler, Richard H
    This is the first of a series of papers on the long-time behavior of [math] –dimensional Ricci flows with surgery. We first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman’s long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman’s techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed.

  10. Long-time behavior of $3$–dimensional Ricci flow: introduction

    Bamler, Richard H
    In the following series of papers we analyze the long-time behavior of [math] –dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by [math] . This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as [math] .

  11. Detecting periodic elements in higher topological Hochschild homology

    Veen, Torleif
    Given a commutative ring spectrum [math] , let [math] be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime [math] , we calculate [math] and [math] for [math] , and use these results to deduce that [math] in the [math] connective Morava K-theory of [math] is nonzero and detected in the homotopy fixed-point spectral sequence by an explicit element, whose class we name the Rognes class. ¶ To facilitate these calculations, we introduce multifold Hopf algebras. Each axis circle in [math] gives rise to a Hopf algebra structure on [math] , and the way these Hopf algebra structures...

  12. The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

    Oblomkov, Alexei; Rasmussen, Jacob; Shende, Vivek
    We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting [math] . By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a [math] torus knot as a certain sum over diagrams. ¶...

  13. Describing the universal cover of a noncompact limit

    Ennis, John; Wei, Guofang
    Suppose that [math] is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds [math] with a uniform lower bound on Ricci curvature. In a previous paper the authors showed that when [math] is compact the universal cover [math] is a quotient of the Gromov–Hausdorff limit of the universal covers [math] . This is not true when [math] is noncompact. In this paper we introduce the notion of pseudo-nullhomotopic loops and give a description of the universal cover of a noncompact limit space in terms of the covering spaces of balls of increasing size in the sequence.

  14. Algebraic and geometric convergence of discrete representations into $\mathrm{PSL}_2\mathbb{C}$

    Biringer, Ian; Souto, Juan
    Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into [math] does not contain parabolics, then it is also the sequence’s geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences of unfaithful representations, and offer a suitable replacement.

  15. Circle homeomorphisms and shears

    Šarić, Dragomir
    We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

  16. Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

    Sorrentino, Alfonso; Viterbo, Claude
    In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather’s [math] function, thus providing a negative answer to a question asked by Siburg [Duke Math. J. 92 (1998) 295-319]. However, we show that equality holds if one considers the asymptotic distance defined in Viterbo [Math. Ann. 292 (1992) 685-710].

  17. Adams operations in smooth $K$–theory

    Bunke, Ulrich
    We show that the Adams operation [math] , [math] , in complex [math] –theory lifts to an operation [math] in smooth [math] –theory. If [math] is a [math] –oriented vector bundle with Thom isomorphism [math] , then there is a characteristic class [math] such that [math] in [math] for all [math] . We lift this class to a [math] –valued characteristic class for real vector bundles with geometric [math] –structures. ¶ If [math] is a [math] –oriented proper submersion, then for all [math] we have [math] in [math] , where [math] is the stable [math] –oriented normal bundle of [math] ....

  18. Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

    Gompf, Robert E; Scharlemann, Martin; Thompson, Abigail
    If there are any [math] –component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. ¶ The simplest plausible counterexample to the Generalized Property R Conjecture could be a [math] –component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to [math] . We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not...

  19. Correction to “A cartesian presentation of weak $n$–categories”

    Rezk, Charles
    This paper gives a corrected proof for Proposition 6.6 of “A Cartesian presentation of weak [math] –categories” [Geom. Topol. 14 (2010) 521–571] by the author.

  20. On the classification of gradient Ricci solitons

    Petersen, Peter; Wylie, William
    We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones [math] , [math] and [math] . This gives a new proof of the Hamilton–Ivey–Perelman classification of [math] –dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of [math] , [math] , [math] , [math] or [math] .

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