Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
New proof of index theorem for topological manifold bundles.
problem Index theorem for fiber bundles of compact topological manifolds.
method Use of a convenient framework for bivariant theories and recent results on the homotopy type of the topological cobordism category.
result Refinement of the assembly map for an extended A-theory characteristic.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length co…
Kontsevich's classes distinguish smooth structures on fiber bundles.
problem Distinguishing smooth structures on fiber bundles.
method Using Kontsevich's characteristic classes and real blow-up construction.
result Kontsevich's classes are determined by the topology of the 2-point configuration space bundle.
The study classifies manifolds that can be split into two disk bundles.
problem Understanding manifolds that can be decomposed into two disk bundles.
method Established through topological restrictions and rational ellipticity.
result Classification of manifolds up to diffeomorphism in dimensions five and six.
The paper proves metrizability and dynamics of Weil bundles.
problem Metrizability and dynamics of Weil bundles in differential geometry.
method Investigation of metrizability and dynamics of Weil bundles for smooth compact manifolds and Weil algebras.
result A canonical, complete, weighted metric \(\mathfrak{d}_w\) on \(M^\mathbf{A}\) that encodes geometry and deformations.
Proves 3-manifold groups uniquely identify hyperbolic bundles.
problem Identifying hyperbolic 3-manifold groups from their finite quotients.
method Upgraded Liu's result to detect fiber type via profinite completion.
result Proves hyperbolic bundles are distinguished by their profinite completions.
This article reviews ∞-bundles and their applications in geometry and physics.
problem Understanding higher bundles in geometry and physics.
method An ∞-categorical formulation of higher bundles. result Identification of higher bundles in various contexts.
Geometric analysis on real analytic manifolds using seminorms.
problem Characterizing operations on real analytic manifolds and vector bundles.
method Using seminorms and geometric decompositions of jet bundles.
result New characterizations of real analytic mappings and operations.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
Study Riemannian metric bundles and their connections to K-theory.
problem Understanding geometry and topology of manifolds with Riemannian metrics.
method Develop rigorous theory of Riemannian metric bundles and apply to K-theory.
result Contribute to deeper understanding of manifold geometry and topology.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
Computes structure set of manifolds homotopy to torus bundles over lens spaces.
problem Computing the structure set of manifolds homotopy equivalent to certain torus bundles.
method Computes topological simple structure set using surgery invariants.
result First computation of structure set for manifolds with non-trivial fundamental group.
It is known that, for Dirac operators on Riemann surfaces twisted by line bundles with Hermitian-Einstein connections, it is possible to obtain estimates for the first eigenvalue in terms of the topology of the twisting bundle \cite{JL2}. Attempts to generalize topological estimates for higher rank bundles or higher di…
The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.
problem Understanding the geometric effects of hyperbolic cohomology classes on Kähler manifolds.
method Introducing Kähler topologically hyperbolic manifolds and proving spectral gap theorems for positive holomorphic Hermitian vector bundles.
result Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class.
We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.
We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.
Given a principal bundle over a closed manifold, G --> P --> M, let P^{Ad} --> M be the associated adjoint bundle. Gruher and Salvatore showed that the Thom spectrum (P^{Ad})^{-TM} is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as th…
New exotic 4-manifolds found with fiber bundles.
problem Finding non-diffeomorphic exotic 4-manifolds.
method Smooth fiber bundles over the circle.
result Exotic 4-manifolds with non-simply-connected fiber.
Paper defines almost flat relative bundles and their correspondence with quasi-representations.
problem Understanding the equivalence of topological and smooth almost flatness.
method Introduces almost flatness for relative bundles, proves equivalence, and defines correspondence with quasi-representations.
result Equivalence of topological and smooth almost flatness and construction of almost flat bundles.
The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.
problem Conditions for self-covering manifolds to be fiber bundles over a circle.
method Developed a criterion for finite generation of modules over a commutative Noetherian ring and proved conditions for fiber bundles.
result Proved that certain self-covering manifolds with free abelian fundamental group are fiber bundles over a circle.
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
Fiber bundles over hyperbolic manifolds with Anosov representations.
problem Understanding the topology of fiber bundles associated with Anosov representations.
method Analyzing the structure of fiber bundles over specific hyperbolic manifolds.
result Fiber bundles over certain hyperbolic manifolds are always smooth fiber bundles.
New configuration space integrals show nontrivial formal smooth structures on 4-manifold bundles.
problem Disproving the 4-dimensional Smale conjecture by constructing nontrivial bundles.
method Defining new configuration space integrals relying on formal smooth structures.
result Discovering a generalized Miller-Morita-Mumford class obstructing formal smooth structures.
Proves unique symplectic Lefschetz fibration from Morse functions.
problem Mapping Morse functions to symplectic Lefschetz fibrations.
method Homotopically unique complex-valued symplectic Lefschetz fibration on cotangent bundles.
result Existence and uniqueness of symplectic Lefschetz fibrations.
The paper studies the topology of hyperspaces of k-dimensional convex sets.
problem Topology of hyperspaces of k-dimensional closed convex sets.
method Proved that hyperspaces are Hilbert cube manifolds with fiber bundle structure over Grassmann manifold.
result Fiber of Kk,bn is homeomorphic with R2k(k+1)+2nimesQ. We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E …
In this article we consider the continuity of the eigenvalues of the connection Laplacian of G-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically G-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric G-…
Study quantizes topological numbers on degenerating Einstein manifolds.
problem Quantizing topological numbers on non-collapsed degenerating Einstein manifolds.
method Compactness theory of bubbles, classical vanishing theorems, and Hirzebruch-Riemann-Roch theorems.
result Established quantization results for various topological numbers.
Researchers calculate the Ray-Singer Torsion for S1 bundles.
problem Few explicit evaluations of path integrals in higher dimensions.
method Algebraic choice of gauge leading to factorization of path integral.
result Explicit calculation of Ray-Singer Torsion for S1 bundles. Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend…
We present a new infinite class of near-horizon geometries of degenerate horizons, satisfying Einstein's equations for all odd dimensions greater than five. The symmetry and topology of these solutions is compatible with those of black holes. The simplest examples give horizons of spatial topology S^3xS^2 or the non-tr…
In this paper we investigate what kind of manifolds arise as the total spaces of iterated S1-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated S1-bundle. We show that the total space of an iterated S1-bundle is homeomorphic to an infra-nilmanifold. A real Bo…
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
The paper studies regular contact manifolds and their products.
problem Characterizing regular contact manifolds and their products.
method Elementary proof for compact manifolds, topological tools for general manifolds.
result Regular contact manifolds are principal bundles with S1 or R structure group. Uniqueness proof for Calderón's problem on real-analytic vector bundles.
problem Recovering topology and geometry from boundary measurements of vector bundles.
method Real-analyticity assumption for uniqueness proof.
result Topology and geometry of vector bundles can be recovered from boundary measurements.
This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let X=G/H be a homogeneous manifold of a Lie group G and let d be a geodesic …
In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundl…
The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing rep…
The abstract discusses the equivalence of transnormal and isoparametric functions on compact manifolds.
problem The existence of transnormal and isoparametric functions on compact manifolds.
method Exploring embedded transnormal systems and showing the existence of transnormal functions on Riemannian manifolds.
result Compact manifolds with transnormal functions also have isoparametric functions, and vice versa.
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.