arXiv research

Survey on foliations and diffeomorphism groups.

The study limits the number of specific foliations with bounded geometry.

Conference compiles problems on foliations and diffeomorphisms.

It is well-known that any isotopically connected diffeomorphism group $G$ of a manifold determines uniquely a singular foliation $\F_G$. A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that…

The paper studies diffeomorphisms of a specific foliation on a Klein bottle.

Study partially hyperbolic diffeomorphisms in 3D, focusing on foliations and dynamics.

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit…

Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.

Study shows how certain foliations in unit tangent bundles behave.

Contractible diffeomorphism groups on lens spaces derived from Morse-Bott foliations.

Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.

The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…

Research proves the connectedness of classifying spaces for Haefliger structures.

Simplified proof of Cerf's theorem on 3-sphere diffeomorphisms.

Closed 4-manifolds foliated by hyperplanes are homeomorphic to the 4-torus.

Regular neighborhoods of singular submanifolds are isotopic to bundle morphisms.

In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification probl…

Anosov flow found in specific partially hyperbolic systems.

Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center…

Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.

Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.

Classifies 3D partially hyperbolic systems, proving ergodicity.

Let V be a representation space of a finite group G. We determine the group structure of the first homology of the equivariant diffeomorphism group of V. Then we can apply it to the calculation of the first homology of the corresponding automorphism groups of smooth orbifolds, compact Hausdorff foliations, codimension …

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus whi…

Study shows Hamiltonian diffeomorphisms form a connected component in $C^0$-topology for most symplectic rational surfaces.

We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we u…

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give an alternative proof of the (surprising…

We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and …

Classifies symmetries of non-flat 3-webs around a point.

It is proven that the identity component of the group preserving the leaves of a generalized foliation is perfect. This shows that a well-known simplicity theorem on the diffeomorphism group extends to the nontransitive case.

Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.

Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an applicat…

Study constructs transverse metrics using transformations commuting with elliptic operators.

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an…

Decomposes flows with jumps into simpler components.

Let $f:M\to M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.

Survey and extend work on singular foliations in diffeology.

According to a theorem of Eliashberg and Thurston a $C^2$-foliation on a closed 3-manifold can be $C^0$-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain non-diffeomorphic contact structures. In this paper …

Sharp spectral estimates for negatively curved foliations.

A rigid submanifold result in contact geometry.

Paper discusses conditions for global injectivity of semi-algebraic local diffeomorphisms.

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology o…

Maximal dilatation found on nonorientable surfaces.

Paper characterizes foliated bundle classes via quasi-morphisms and studies their boundedness.

In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an…