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Emmanuel Giroux

Publications and source records attributed to Emmanuel Giroux.

13 recordsLinked to original sources
arXiv2025

From Morse Functions to Lefschetz Fibrations on Cotangent Bundles

We prove that, for any Morse function on a compact manifold and any adapted gradient satisfying the Morse-Smale condition, there is a homotopically unique complex-valued symplectic Lefschetz fibration on the cotangent bundle whose restriction to the zero-section is the given function, whose imaginary part is the evaluation of covectors on the gradient, and which is equivariant under the actions of the fiberwise antipodal involution and the complex conjugation. Then we study the topology and symplectic geometry of the regular fibers of this fibration, which are well-defined Weinstein manifolds.

math.SG↗
arXiv2018

Remarks on Donaldson's symplectic submanifolds

This paper presents a few remarks about the topology of symplectic hyperplane sections and the geometry of their complements. In particular, it contains a detailed proof of the following result already stated with hints in [Gi]: for sufficiently large degrees, the complements of Donaldson's symplectic hyperplane sections are naturally Weinstein --- and less naturally Stein --- manifolds.

math.SG↗
arXiv2017

Ideal Liouville Domains - a cool gadget

Liouville domains have become central objects in symplectic and contact geometry. However, the auxiliary data they involve --- namely, Liouville forms --- and the non-compactness of their completions generate some inconvenience. The notion of ideal Liouville domains is designed to suppress these awkward aspects and to let symplectic structures play the leading role. Ideal Liouville domains are compact manifolds with boundary whose interior carries a symplectic form satisfying some tameness condition along the boundary. Their definition and their basic properties are presented in the first part of these notes, while the second part discusses their relevance in contact geometry.

math.SG↗
arXiv2015

On the contact mapping class group of Legendrian circle bundles

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.

math.GT↗
arXiv2009

On the stable equivalence of open books in three-manifolds

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the theorem implies a conjecture of J Harer, namely, that any fibered link in the three-sphere can be obtained from the unknot by a sequence of plumbings and deplumbings of Hopf bands. The proof presented here involves contact geometry in an essential way.

math.GT↗
arXiv2003

Notes on the isotopy finiteness

This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

math.GT↗
arXiv2003

On the coarse classification of tight contact structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

math.GT↗
arXiv2003

Géométrie de contact: de la dimension trois vers les dimensions supérieures

On décrit ici des relations entre la géométrie globale des variétés de contact closes et celle de certaines variétés symplectiques, à savoir les variétés de Stein compactes. L'origine de ces relations est l'existence de livres ouverts adaptés aux structures de contact. We discuss relations between the global geometry of closed contact manifolds and the geometry of compact symplectic Stein manifolds that they bound. The origin of these relations is the existence of open book decompositions adapted to contact structures.

math.GT↗
arXiv2001

Sur les transformations de contact au-dessus des surfaces

Let S be a compact surface - or the interior of a compact surface - and let V be the manifold of cooriented contact elements of S equiped with its canonical contact structure. A diffeomorphism of V that preserves the contact structure and its coorientation is called a contact transformation over S. We prove the following results. 1) If S is neither a sphere nor a torus then the inclusion of the diffeomorphism group of S into the contact transformation group is 0-connected. 2) If S is a sphere then the contact transformation group is connected. 3) if S is a torus then the homomorphism from the contact transformation group of S to the automorphism group of $H_1(V) \simeq Z^3$ has connected fibers and the image is (known to be) the stabilizer of $Z^2 \times \{0\}$).

math.GT↗
arXiv1999

Structures de contact sur les varietes fibrees en cercles au-dessus d'une surface

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isotopic to a contact structure transverse to the fibers or there exists, in some finite sheeted cover of V, a Legendrian curve isotopic to the fiber along which $ξ$ determines the same framing as the fibration $V \to S$. In Section 3, we classify contact structures that are transverse to the fibers up to isotopy and conjugation. In Section 4, we study general tight contact structures on V. We prove that virtually over-twisted contact structures form finitely many isotoy classes while isotopy classes of universally tight contact structures are in one-to-one correspondence with isotopy classes of systems of essential curves on S.

math.GT↗
arXiv1999

Structures de contact en dimension trois et bifurcations des feuilletages de surfaces

The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above.

math.GT↗