Contact homology and homotopy groups of the space of contact structures

Bourgeois, Frédéric

doi:10.4310/mrl.2006.v13.n1.a6

Cited by 39 publications

(56 citation statements)

References 12 publications

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“…Arguing as in Lemma 4.4 we see that S ∞ ∩ c E has one or two connected components which are punctured spheres. Since S ∞ is a topological sphere, we conclude that 5 at the centres of these balls, the area of any J ∞ -curve passing through p 1 , . .…”

Section: Theorem 5 the Space Of Symplectic Embeddings Of Ellipsoidsmentioning

confidence: 88%

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Polarizations and symplectic isotopies

Opshtein

2013

Journal of Symplectic Geometry

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The aim of this paper is to explain a link between symplectic isotopies of open objects such as balls and flexibility properties of symplectic hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids or maximal packings of P 2 . IntroductionIn [3], Biran proved a decomposition theorem for rational Kähler manifolds which has proved useful in many situations such as symplectic packings [2, 20], Lagrangian embeddings [4] . . . This paper tries to add symplectic isotopies to this list of applications. In polarized symplectic manifolds -triples (M, ω, Σ) where ω ∈ H 2 (M, Z) and Σ is a symplectic hypersurface Poincaré-dual to a (necessarily positive) multiple kω of the symplectic form -, this decomposition result may be expressed as:Theorem 1 (Biran). In a polarized closed symplectic manifold (M, ω, Σ), there exists a zero-volume closed skeleton in M whose complement is a standard symplectic disc bundle supported by Σ.In fact, polarizations of sufficiently high degree exist on all closed (i.e., compact, without boundary) rational symplectic manifolds [7]. These manifolds split into a standard symplectic part -an explicit disc bundle over Σ -and a negligible skeleton, even isotropic in the Kähler case. Although no kind of uniqueness can be expected (see [3] for a discussion on the skeleton for instance), our next theorem explains how to construct many such decompositions, all of whose symplectic parts are isotopic. In order to state it, let us mention that both M \Σ and the complement of the zero-section L 0 in the symplectic disc bundle SDB(Σ, k) are exact symplectic manifolds, so they admit Liouville forms. In the statement below, λ 0 is a distinguished such form on SDB(Σ, k)\L 0 (see Section 1). Also, by a local embedding of (X, M ) into (Y, N ) -where X ⊂ M and Y ⊂ N -we always mean an 109

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Section: Theorem 5 the Space Of Symplectic Embeddings Of Ellipsoidsmentioning

confidence: 88%

“…We then have (see [9], Proposition 1.7.1 for the dimension of the moduli spaces and [5,8] for the regularity assertions):…”

Section: Sft and Supporting Surfacesmentioning

confidence: 99%

Polarizations and symplectic isotopies

Opshtein

2013

Journal of Symplectic Geometry

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“…There is also an obvious relation between Lagrangian concordance and Legendrian contact homology. Following Bourgeois [5] we see that a Lagrangian cylinder between two Legendrian knots could be used to define a map between the algebras CH .K C / and CH .K / (see Ekholm, Etnyre and Sullivan [6]). We, however, will not give a more detailed description of this map for two reasons.…”

Section: Discussionmentioning

confidence: 99%

Lagrangian concordance of Legendrian knots

Chantraine

2010

Algebr. Geom. Topol.

116

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In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Thom conjecture.Comment: 18 pages, 4 figures. v2: Minor corrections and a proof of conjecture 6.4 of version 1 (now Theorem 6.4). v3: Several substantial changes notably the proof of theorems 1.2 and 5.1, this is the version accepted for publication in "Algebraic and Geometric Topology" published in January 201

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“…As already demonstrated by Seidel [26] and Bourgeois [4], the continuation (monodromy) map can prove non-triviality theorems, in a way analogous to the above, about the fundamental group of the space of objects ("object" means a symplectomorphism in [26] and a contact structure in [4]), to which a version of Floer theory is associated 3 . To get such conclusions, it is of course a key point to prove that monodromy is invariant under homotopies of the loop of objects.…”

Section: Floer Theorymentioning

confidence: 95%

Contact homology and one parameter families of Legendrian knots

Kálmán

2005

Geom. Topol.

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We consider S 1 -families of Legendrian knots in the standard contact R 3 . We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop (Theorem 1.1). We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S 1 , R 3 ) of Legendrian knots, although it is contractible in the space Emb(S 1 , R 3 ) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid (Definition 6.1) and construct an augmentation for each such link diagram.

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Contact homology and homotopy groups of the space of contact structures

Abstract: Abstract. Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higherdimensional contact manifolds and higher order homotopy groups.

Cited by 39 publications

References 12 publications

Polarizations and symplectic isotopies

Polarizations and symplectic isotopies

Lagrangian concordance of Legendrian knots

Contact homology and one parameter families of Legendrian knots

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