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.

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras-Russell cubic is Stein deformation equivalent to C 3 and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.

We present the first examples of elements in the fundamental group of the space of Legendrian links in false(S3,ξstfalse)$(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer‐theoretic techniques. These new families include the first‐known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer‐theoretic proof that Legendrian false(n,mfalse)$(n,m)$ torus links have infinitely many Lagrangian fillings if n⩾3,m⩾6$n\geqslant 3,m\geqslant 6$ or false(n,mfalse)=false(4,4false),false(4,5false)$(n,m)=(4,4),(4,5)$. In addition, for any given higher genus, we construct a Weinstein 4‐manifold homotopic to the 2‐sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.

Abstract. 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.

In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higherdimensional Legendrian submanifolds.