We give necessary and sufficient conditions for a closed orientable 9-manifold M to admit an almost contact structure. The conditions are stated in terms of the Stiefel-Whitney classes of M and other more subtle homotopy invariants of M.
By a fundamental result of Borman, Eliashberg and Murphy, M admits an almost contact structure if and only if M admits an over-twisted contact structure. Hence we give necessary and sufficient conditions for M to admit an over-twisted contact structure and we prove that if N is another closed 9-manifold which is homotopy equivalent to M, then M admits an over-twisted contact structure if and only if N does.
In addition, for W_i(M) the i-th integral Stiefel-Whitney class of M, we prove that if W_3(M) = 0 then W_7(M) = 0.
We construct new examples of contact manifolds in arbitrarily large dimensions. These manifolds which we call quasi moment-angle manifolds, are closely related to the classical moment-angle manifolds.
A K-contact manifold is a smooth manifold M with a contact form whose Reeb flow preserves a Riemannian metric on M. Main examples are Sasakian manifolds. Our results in this paper are four results i), ii), iii) and iv) below obtained by the application of the Morse theory to the contact moment maps on closed 5-dimensional K-contact manifolds of rank 2. 5-dimensional K-contact manifolds of rank 2 have the lowest dimension among K-contact manifolds with nontrivial contact moment maps which are not toric. i) Correspondence between the isomorphism classes of K-contact manifolds and graphs of isotr…
We prove that on overtwisted contact manifolds there can be no positive loops of contactomorphisms that are generated by a $C^0$-small Hamiltonian function.
We prove some contact analogs of smooth embedding theorems for closed $π$-manifolds. We show that a closed, $k$-connected, $π$-manifold of dimension (2n + 1) that bounds a $π$-manifold, contact embeds in the $(4n-2k+3)$-dimensional Euclidean space with the standard contact structure. We also prove some isocontact embedding results for $π$-manifolds and parallelizable manifolds.
Computing embedded contact homology (ECH) and related invariants of certain toric 3-manifolds (in the sense of Lerman) has led to interesting new results in the study of symplectic embeddings. Here, we give a combinatorial formulation of ECH chain complexes for general toric contact 3-manifolds. As a corollary, we prove Conjecture A.3 from [Hutchings14].
We prove some open book embedding results in the contact category. For example, we show that a large class of contact 3-manifolds admit contact open book embedding in the standard contact 5- sphere. We also prove that all the Ustilovsky (4m + 1)-spheres contact open book embed in the standard contact (4m + 3)-sphere.
We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold $(Σ,ξ=\ker(α))$. Previous results show that a large class of Liouville-fillable contact manifolds admit contractible positive loops. In contrast, we show that for any Liouville-fillable $(Σ,α)$ with $\dim(Σ) \geq 7$, there exists a Liouville-fillable contact structure $ξ'$ on $Σ$ which admits no positive loop at all. Further, $ξ'$ can be chosen to agree with $ξ$ on the complement of a Darboux ball.
Let $(M, α)$ be a $2n+1$-dimensional connected compact contact toric manifold of Reeb type. Suppose the contact form $α$ is regular, we find conditions under which $M$ is homeomorphic to $S^{2n+1}$.
In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
In this paper we study embeddings of contact manifolds using braidings of one manifold about another. In particular we show how to embed many contact 3-manifolds into the standard contact 5-sphere. We also show how to obstruct braidings of one manifold about another using contact geometry.
A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold, and use this to prove versions of conjectures of Sandon and Mazzucchelli on the existence of translated points and invariant Reeb orbits, and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.
We give an algorithm for computing the contact homology of some Brieskorn manifolds. As an application, we construct infinitely many contact structures on the class of simply connected contact manifolds that admit nice contact forms (i.e. no Reeb orbits of degree -1,0 or 1) and have index positivity with trivial first Chern class.
We show that certain classes of contact 3-manifolds do not admit non-separating contact type embeddings into any closed symplectic 4-manifolds, e.g. this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic 4-manifolds. Similarly, there are symplectic 4-manifolds that can admit smoothly embedded non-separating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.
We prove that closed connected contact manifolds of dimension $\geq 5$ related by an h-cobordism with a flexible Weinstein structure become contactomorphic after some kind of stabilization. We also provide examples of non-conjugate contact structures on a closed manifold with exact symplectomorphic symplectizations.
We introduce the notion of coLegendrian submanifold in a contact manifold as a middle-dimensional coisotropic submanifold and study its existence in dimension $5$ via contact Morse theory. As an application, the second generation of Orange is now available.
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifoldswith singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists.
We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2χ+3σ$ for all its minimal symplectic fillings. This is motivated by the bound of Stipsicz for Stein fillings. Finally, we discuss various aspects of exact self cobordisms of fillable 3-manifolds.
Using the structural theorems developed in [Hua13], we study the deformation theory of coisotropic submanifolds in contact manifolds, under the assumption that the characteristic foliation is nonsingular. In the "middle" dimensions, we find an interesting relationship with foliation theory. Some elementary applications of contact geometry in foliation theory are explored.
We express the mean Euler characteristic of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical surgery and examine the behavior of the mean Euler characteristic under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the mean Euler characteristic in the Morse-Bott case.
In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper, we introduce the Cartan prolongation of a contact 3-orbifold and the development map associated to a more general Engel manifold, and we give necessary and sufficient condition for the Cartan prolongation to be a manifold. Moreover, we explain the Cartan prolongation of a 3-dimensional contact étale Lie groupoid and the development map associated to any Engel…
In this paper, we give an open book decomposition for the contact structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux.
These are notes for a course on contact manifolds and torus actions delivered at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems at Centre de Recerca Matemàtica in Barcelona in July 2001. To be published by Birkhauser.
We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact m…
According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $π$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and most of them are strongly symplectically fillable. The proof is based on the Lerman's classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of $T^d, d\geq3$ $(T…
Infinitely many contact 3-manifolds each admitting infinitely many, pairwise non-diffeomorphic Stein fillings are constructed. We use Lefschetz fibrations in our constructions and compute their first homologies to distinguish the fillings.
We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.
We give a proof of, for the case of contact structures defined by global contact 1-forms, a Theorem stated by Eliashberg that for any overtwisted contact structure on a closed 3-manifold, its contact homology is 0. A different proof is also outlined in the appendix by Yakov Eliashberg.
For a certain class of exotic contact manifolds of dimension greater than 3, we show that there is an abundance of closed exact Lagrangians in their symplectization. All of these Lagrangians are displaceable by Hamiltonian isotopy, and many of the examples are nulhomotopic.
In this note we analyze displaceability of pre-Lagrangian toric fibers in contact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we show that this is not the case for contact toric manifolds. More precisely, we prove that for the contact toric manifolds $\mathbb{S}^{2d-1} (d\geq 2)$ and $\mathbb{T}^k \times \mathbb{S}^{2d+k-1} (d \geq 1)$ all pre-Lagrangian toric fibers are displaceable, and that for all contact toric manifolds for which the toric action is free, except possibly non-tri…
In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure ξ_- that also contains a plastikstufe. As a consequence, every closed contact manifold M (except S^1) can be converted into a contact manifold that is not (semi-positively) fillable by taking the connected sum of M with (S^{2n-1},ξ_-).
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.
The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, ξ_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, ξ_N),$ provided $M$ contact embeds in $(N, ξ_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $ξ_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sp…
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to a weakly convex good cone by the cutting construction. (Note that these toric contact manifolds can not be constructed by Delzant construction.) We further prove there are no toric Sasakian structures on these contact manifolds. From this, contact toric manifolds of toric K-cont…
We show that, under a certain condition, contact 5-manifolds can `coarsely' distinguish smooth structures on compact Stein 4-manifolds via contact open books. We also give a simple sufficient condition for an infinite family of Stein 4-manifolds to have an infinite subfamily of pairwise non-diffeomorphic Stein 4-manifolds. The proofs rely on the adjunction inequality. We remark that there are various examples of infinitely many pairwise exotic Stein 4-manifolds whose smooth structures can be distinguished by these results.
We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.
According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterise the standard contact structure on the 5-sphere in terms of such fillings. This characterisation is discussed in the context of the Andrews-Curtis conjecture concern…
We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.
We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.
It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.
In this paper, we show the existence of (co-oriented) contact structures on certain classes of $G_2$-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with $G_2$-structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with $G_2$-structures.
In this paper, we find infinite hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that part of these manifolds do admit tight contact structures.
We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on $S^3$ preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by $S^3$ to the exceptional isomorphism $SO(4)=(SU(2)\times SU(2))/{\pm 1}$. The main tool used is equivariant framings of 3-manifolds.
We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every orientable three-submanifold Y of an orientable seven-manifold with spin structure such that for every co-oriented contact structure on Y, N admits a co-oriented contact structure such that Y is a contact submanifold of N. Moreover, we construct infinitely many pairwise distinct…
We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.
We construct symplectic submanifolds of symplectic manifolds with contact border. The boundary of such submanifolds is shown to be a contact submanifold of the contact border. We also give a topological characterization of the constructed submanifolds by means of a ``relative Lefschetz hyperplane Theorem''. We sketch some of the applications of the results.
Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.
