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The Bourgeois construction associates to every contact open book on a manifold $V$ a contact structure on $V\times T^2$. We study in this article some of the properties of $V$ that are inherited by $V\times T^2$ and some that are not. Giroux has provided recently a suitable framework to work with contact open books. In the appendix of this article, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this new language., 25 pages; reorganized structure of paper, addressed Gironella's work on Bourgeois structures, improved exposition and added examples

In this paper we prove the presence of an embedded plastikstufe implies overtwistedness of the contact structure in any dimension. Moreover, we show in dimension 5 that the presence of an embedded bordered Legendrian open book (bLob) also implies overtwistedness. Applications of these criteria to certain (contact) fibered connected sums and open books are discussed., Comment: final version; substantial revisions; to appear in the Journal of Topology

Let $L$ be a closed totally real submanifold of $\mathbb{C}^{n}$, $n\ge 2$, which is not Lagrangian. We observe that small enough tubular neighborhoods of $L$ give exotic examples of weak fillings of $ST^{\ast}L$ endowed with its standard contact structure., Comment: 10 pages. The referee's comments have been included

The aim of this text is to give an accessible overview to some recent results concerning contact manifolds and their symplectic fillings. In particular, we work out the weakest compatibility conditions between a symplectic manifold and a contact structure on its boundary to still be able to obtain a sensible theory (Chapter II), furthermore we prove two results (Theorem A and B in Section I.4) that show how certain submanifolds inside a contact manifold obstruct the existence of a symplectic filling or influence its topology. We conclude by giving several constructions of contact manifolds that for different reasons do not admit a symplectic filling.

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

This paper proves that two possible notions of Stein fillability for a contact structure are the same. A contact geometer typically says a contact manifold $(M,\xi)$(M,ξ) is Stein fillable if $M$M is the noncritical level set of a proper plurisubharmonic function $f\colon X\to [0,\infty)$f:X→[0,∞) and $\xi$ξ is the set of complex tangencies to the level set $M.$M. This is a very useful notion in contact geometry and implies strong geometric properties for the contact structure if $M$M is 3-dimensional. (In particular, the contact structure will be tight. Moreover, many tight contact structures are constructed by constructing a Stein filling of a 3-manifold.) There is another, "more intrinsic'', notion of Stein fillability. Suppose $X$X is a Stein manifold and $X$X is the interior of a closed manifold $\overline{X}$X−− such that $\overline{X}=M\cup X$X−−=M∪X and the complex structure on $X$X extends to one on $\overline{X}. $X−−. Then we can say the contact manifold $(M,\xi)$(M,ξ) is filled by $X$X, where $\xi$ξ is the set of complex tangencies to $M.$M. In this case the authors say that $M$M is the "intrinsic'' boundary of $X.$X. The main result of this paper shows the equivalence of these two notions. The first notion obviously implies the second, but the converse is non-obvious. Along the way the authors prove several results. In particular, they show that if $(M,\xi)$(M,ξ) is a 3-dimensional Stein fillable contact manifold then $M$M embeds in complex 4-space such that $\xi$ξ is the set of complex tangencies to the embedding. They also show that if $M$M is an intrinsic boundary of $X$X then $\overline{X}$X−− embeds as a domain in a larger open complex manifold with strictly pseudoconvex boundary $M.$M. However, the germ of this extension of $\overline{X}$X−− is not unique.