We provide examples of contact manifolds of any odd dimension $\geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations

The nearby Lagrangian conjecture predicts that any closed exact Lagrangian submanifold L in the cotangent bundle T * M of a closed manifold M is Hamiltonian isotopic to the zero-section; we refer to such a submanifold L as a nearby Lagrangian. The conjecture is wide open in general and presently known to hold only for M = S 1 (where it is elementary), M = S 2 ([Hin12]) and M = T 2 ([DRGI16]). A first obstruction is that the projection π : L → M may not even be homotopic to a diffeomorphism, or even worse that L may not be diffeomorphic to M. The first and fourth authors have shown in [Kra13] and [Abo12b] that π is at least a homotopy equivalence, a result reproved by the third author in [Gui12]. In [AK18], the first and fourth authors further proved that π is a simple homotopy equivalence. Constraints on the smooth structure of L have also been discovered, e.g. by the first author in [Abo12a] in the case where M is a sphere (see also [EKS15]).We do not pursue this line of research in this paper, but instead focus on the following: if the conjecture holds, then the tangent bundle of L and the cotangent fibres define homotopic sections of the restriction of the Lagrangian Grassmannian bundle of T * M to L. This implies the triviality of the stable Gauss map L → Λ 0 (C ∞ ) (see Definition 2.5, the stable Lagrangian Grassmannian U/O is denoted Λ 0 (C ∞ ) in this paper). Our main result in this regard is the following: Theorem A. Let M be a closed manifold and L a closed exact Lagrangian submanifold of T * M, then the stable Gauss map L → Λ 0 (C ∞ ) vanishes on all homotopy groups.

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor-Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration.The group of compactly supported diffeomorphisms of Euclidean space R 2m is denoted by Diff c (R 2m ). It is equipped with the compact-open topology; its set of connected components π 0 Diff c (R 2m ) inherits a group structure, which coincides with the group of exotic (2m + 1)dimensional spheres under connected sum. Given a mapping class η ∈ Diff c (R 2m ), we denote by Σ η ∈ Θ 2m+1 the corresponding exotic sphere. 2.1. Smooth Milnor-Munkres pairing. The Milnor-Munkres pairing is, in its simplest form [21, p. 583], a group homomorphism

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold (V, ξ) has a supporting open book whose pages are flexible Weinstein manifolds, then the connected component G of the identity in its automorphism group is a uniformly simple group: for every non-trivial element g, every other element is a product of at most 128(dim V + 1) conjugates of g ±1 . In particular any conjugation invariant norm on this group is bounded. We also prove the later statement still holds for the universal cover of G.

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