Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles fo…
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48 results for “foliated ρ-classes”
The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions satisfying certain con…
This paper extends foliation concepts to singular foliations using Lie -algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie -algebroids to define modular class.
result Geometric meaning of modular class as an obstruction to universal Lie -algebroids.
We construct the equivalent of the Godbillon-Vey class and its generalizations for regular foliations on super-manifolds. We interpret these classes as classes of foliated flat connections.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.
problem Finding higher-dimensional analogs of the Calabi invariant and its transgression to the Euler class.
method Constructing a cohomology class of volume-preserving diffeomorphisms and proving transgression to the Euler class of foliated sphere bundles.
result The cohomology class transgresses to the Euler class of foliated sphere bundles.
This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
problem Proving a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
method Analyzing the Euler class and using hyperbolic geometry properties.
result For a taut foliation on a hyperbolic 3-manifold, if the Euler characteristic of a closed leaf equals the Euler class, then there exists another taut foliation with the same Euler class.
A formula calculates the Euler class of foliations using dual graphs.
problem Calculating the Euler class of foliations using cooriented branched surfaces.
method Using dual graphs of cooriented branched surfaces to define a simplicial 1-cycle representing the Poincaré dual of the Euler class.
result The formula generalizes previous results and classifies realizable homology classes.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…
The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
problem Constructing 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
method Constructs infinitely many rational homology 3-spheres using Dehn surgeries and Heegaard Floer homology.
result Found rational homology 3-spheres that admit co-orientable taut foliations but none with vanishing Euler class.
Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for…
Classifying foliationsmath.DS
We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth …
This paper extends NCFI to odd codimension and computes examples.
problem Extending NCFI to foliations of odd codimension.
method Computing NCFI for various foliated manifolds in both even and odd codimensions.
result NCFI is an invariant of foliations in odd codimension, requiring an odd \(K_1\)-class.
New evidence supports the Euler class one conjecture for tight contact structures.
problem Euler class one conjecture for taut foliations and tight contact structures.
method Analysis of tight contact structures and counterexamples to the conjecture.
result Counterexamples to the Euler class one conjecture for taut foliations are also Euler classes of tight contact structures.
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
Paper characterizes foliated bundle classes via quasi-morphisms and studies their boundedness.
problem Characterizing bounded characteristic classes of foliated bundles.
method Using non-descendible quasi-morphisms on the universal covering of the structure group.
result Non-existence of foliated structures on some Hamiltonian fibrations and non-triviality of the second bounded cohomology group.
The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.
Introduces positivity for classes in foliated manifolds.
problem Understanding positivity in foliated manifolds.
method Introduces positivity for a real basic class in basic Bott-Chern cohomology group and studies its relationship with negativity of transverse holomorphic sectional curvature.
result Establishes the relationship between positivity and negativity in foliated manifolds.
The interior Kasparov product formula is extended for foliated ρ-classes on Riemannian bundles.
problem Extending the Kasparov product formula for foliated ρ-classes.
method Construction of asymptotic morphisms and adiabatic deformation groupoids.
result Interior Kasparov product formula for foliated ρ-classes on Riemannian bundles.
We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz coh…
The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliationmath.DG
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
Study on characteristic classes for foliation deformations.
problem Characterizing and understanding characteristic classes for foliation deformations.
method Introduced a differential graded algebra (DGA) to recover Bott vanishing and formulae, and discussed properties of its cohomology.
result Discovered new classes that cannot be described by existing classes like Godbillon--Vey and Fuks--Lodder--Kotschick.
We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.
Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.
problem Bounding the dual Thurston norm of foliations on 3-manifolds of negative curvature.
method Uses constants like injectivity radius, volume, curvature, and mean curvature of foliation leaves to estimate the dual Thurston norm.
result Provides an upper bound estimate on the dual Thurston norm of the Euler class of a foliation.
A class of codimension one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally non-degenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson type submanifold. In particular we deduce that their leaf spaces are homeomor…
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on .
method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
Survey on Killing foliations with technical advantages.
problem Understanding closures of Riemannian foliations.
method Review of Molino's structural theory and transverse isometry theory.
result Closures of Killing foliations described by transverse Killing vector fields.
The paper bounds the -norm of Euler class for foliations on 3-manifolds.
problem Bounding the -norm of the Euler class for foliations on 3-manifolds.
method Using constants bounding volume, radius of injectivity, sectional curvature, and mean curvature of leaves.
result Only finitely many cohomological classes can be realized by the Euler class of a transversely oriented foliation with bounded mean curvature.
The aim of the paper is to construct some Godbillon-Vey classes of a family of regular foliations, defined in the paper. These classes are cohomology classes on the manifold or on suitable open subsets. Some examples are also considered.
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of…
Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse…
In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterizat…
Introduces zebra structures on surfaces for directional foliation.
problem Finding canonical representatives of homotopy classes on surfaces.
method Introduces zebra structures and uses triangulations with edge connections to prove existence.
result Canonical representatives exist for homotopy classes on surfaces with specific triangulations.
Godbillon-Vey classes for Lie subalgebroids
problem Extending Godbillon-Vey class to singular foliations
method Defining Godbillon-Vey class for Lie subalgebroids
result Presenting several examples
Approximating -foliationsmath.GT
We extend the Eliashberg-Thurston theorem on approximations of taut oriented -foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented -foliations, where by foliation, we mean a foliation with continuous tangent plane field. These -fol…
Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.
problem Creating foliations of lightcones with specific geometric properties.
method Employing a geometric flow inspired by Huisken-Yau's approach for Riemannian settings.
result Initial data converges exponentially to an STCMC surface under area preserving null mean curvature flow.
Study on complex tori foliations and flat geometries.
problem Understanding turbulent foliations on compact complex tori.
method Defined and analyzed smooth turbulent foliations on compact complex tori.
result All transversely holomorphic Cartan geometries are flat.
We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is simply connected (or more generally if the foliation is a transversely orientable Killing foliation), if M is complete and if the space of leaf closures is compact, then the basic c…
We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equiva…
Defines transverse symbols for foliated manifolds and proves their K-homology class.
problem Transverse index theory for foliated manifolds.
method Using filtrations of tangent bundles, defining transverse symbols, and constructing equivariant KK-classes.
result Transversally Rockland operators yield a K-homology class and there is a Poincare duality result.
Rigidity of the Álvarez classes of Riemannian foliations with nilpotent structure Lie algebrasmath.DG
We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the Álvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the Álvarez class under deformations shown in arXiv:1009.1098v2, …
Study of singular foliations of b^k-type and their geometric properties.
problem Classify and understand singular foliations of b^k-type.
method Introduce and study singular foliations of b^k-type, classify them, and relate them to geometric structures.
result Obstructed by a characteristic class when extending k-th order foliations to (k+1)-th order foliations.
In this article, we study the Euler class of taut foliations on the Dehn fillings of a -homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillin…
Survey and extend work on singular foliations in diffeology.
problem Understanding singular foliations and their properties in diffeological settings.
method Survey Stefan and Sussmann's work, introduce transverse equivalence, and define basic cohomology.
result Transverse equivalence of singular foliations preserves leaf spaces diffeologically but not conversely.
Subject of present paper is the review of results of authors on foliation theory and applications of foliation theory in control systems. The paper consists of two parts. In the first part the results of authors on foliation theory are presented, in the second part the results on applications of foliation theory in the…