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We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at π-level., The authors are supported by Spanish Research Projects SEV?2015?0554, MTM2016?79400?P and MTM2015?72876?EXP. Fernández is supported by a Master?Severo Ochoa grant and by Beca de Personal Investigador en Formación UCM. MartínezAguinaga is funded by Programa Predoctoral de Formación de Personal Investigador No Doctor del Departamento de Educación del Gobierno Vasco.

Kostant gave a model for the geometric quantization via the coho-mology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the co-homology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an in-tegrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme., Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize and is partially supported by grants with reference MTM2015-69135-P (MINECO-FEDER) and 2017SGR932 (AGAUR). Romero Solha is supported by CAPES and partially supported by MTM2015-69135- P (MINECO/FEDER). Francisco Presas is supported by the grant reference number MTM2016-79400-P (MINECO/FEDER). Eva Miranda and Francisco Presas are sup- ported by an EXPLORA CIENCIA project with reference number MTM2015-72876-EXP and by the excellence project SEV-2015-0554

We study homotopically non-trivial spheres of Legendrians in the standard contact \({\mathbb {R}}^3\) and \({\mathbb {S}}^3\). We prove that there is a homotopy injection of the contactomorphism group of \({\mathbb {S}}^3\) into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.

A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $\omega$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, \omega)$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F, F \cap W_k)$, with $F$ any leaf of $\mathcal{F}$. This is applied to draw important consequences on the transverse geometry of such foliations., Comment: Title and abstract modified. Section 2 on Lie groupoids and essential equivalence greatly reduced. bibliography updated. DOI added (to appear in Nagoya Math. J.)

The existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5-dimensional manifold.

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu's result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption., Comment: 2 figures

Algebr. Geom. Topol. 18 (2018) 4275-4303; International audience; We relate open book decompositions of a 4-manifold M with its Engel structures. Our main result is, given an open book decomposition of M whose binding is a collection of 2-tori and whose monodromy preserves a framing of a page, the construction of an En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a contactomorphism. As a consequence, on a parallelizable closed 4-manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that amongst the supported Engel structures we construct, there is a class of loose Engel structures.

Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them. Besides providing their basic properties and presenting two series of examples, we classify those satisfying certain positivity conditions.

In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and the subspace of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds., Comment: 9 pages, 2 figures

The loop space associated to a Riemannian manifold admits a quasisymplectic structure (that is, a closed 2-form which is non-degenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied.

International audience; Given an overtwisted contact manifold, we prove that there can be no positive loops of contactomorphisms that are generated by a C^0-small Hamiltonian function.

We define Maslov-type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of Z . These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these indices and their applications.

A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the presence of an overtwisted leaf. The result is shown to be sharp., 17 pages

In this article we prove that the inclusion of the space of Engel structures of a smooth $4$-fold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. In particular, we construct Engel structures representing any given full flag., 21 pages, 7 figures

Open access version at https://arxiv.org/abs/1503.06221, In this article we establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then we characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, we provide several applications of these geometric criteria., The first author was supported by NSF grant DMS-1841913 and a BBVA Research Fellowship. The second author was supported by NSF grant DMS-1510305 and a Sloan Research Fellowship. The third author was supported by Spanish Research Projects SEV–2015–0554, MTM2016– 79400–P, and MTM2015–72876–EXP.

In this article, we introduce the topological study of codimension-1 foliations which admit contact structures on the leaves. A parametric existence h-principle for foliated contact structures is provided for any cooriented foliation in a closed oriented four-fold.

We define the concept of Lefschetz contact pencil and we show the existence of such structures on any contact manifold. The main idea of the proof is a generalization of the Donaldson arguments used in the symplectic case. We will analyze some of the applications of such existence theorem for the topology of approximately holomorphic contact submanifolds., Comment: 26 pages, 1 figure

These notes are intended to be an introduction to the use of approximately holomorphic techniques in almost contact and contact geometry. We develop the setup of the approximately holomorphic geometry. Once done, we sketch the existence of the two main geometric decompositions available for an almost contact or contact manifold: open books and Lefschetz pencils. The use of the two decompositions for the problem of existence of contact structures is mentioned.

We prove the non--triviality of the Reeb flow for the (2n+1)--dimensional standard contact spheres inside the fundamental group of their contactomorphism group, n greater than 3. The argument uses the existence of homotopically non--trivial 2--spheres in the space of contact structures of a 3--Sasakian manifold., 8 pages

We provide various definitions for the contact blow--up. Such different approaches to the contact blow--up are related. Some uniqueness and non--uniqueness results are also provided., 23 pages, 2 figures. Minor changes made in the last version

The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nx{0} in the model space NxR^{2k}. In this article we make the observation that if (N,\xi_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,\xi), and if its model neighborhood is sufficiently large, then (M,\xi) does not admit an exact symplectic filling., Comment: 19 pages, 2 figures; added example of manifold that is not fillable by neighborhood criterium; typos

A geometric obstruction, the so called "plastikstufe", for a contact structure to not being fillable has been found by K. Niederkruger. This generalizes somehow the concept of overtwisted structure to dimensions higher than 3. This paper elaborates on the theory showing a big number of closed contact manifolds with a "plastikstufe". They are the first examples of non-fillable contact closed high-dimensional manifolds. In particular we show that $S^3 \times \prod_j \Sigma_{j}$, for $g(\Sigma_j)\geq 2$, possesses this kind of contact structure and so any connected sum with those manifolds also does it., Comment: 19 pages, 4 figures

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve., Comment: (one small mistake corrected)

We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologically trivial vanishing spheres in these pencils. Moreover we show some basic topological properties of the space of asymptotically holomorphic transverse sections. These properties allow us to define a new set of symplectic invariants of the original symplectic structure.

We use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r,N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of embeddings in the projective space $CP^N$ obtaining control on N. The main reason of our study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain ``generalized Schur cycles'' defined on a product of grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Auroux as zero sets of approximately holomorphic sections of ``very ample'' vector bundles., 40 pages, no figures, Latex2e

We prove the existence of contact submanifolds realizing the Poincaré dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact category to Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfy a transversality with estimates property directly in the contact category. The description of the obtained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.