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F. Bourgeois, T. Ekholm, Y. Eliashberg, Effect of Legendrian surgery.
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Abstract :
The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown that the results of this paper imply P. Seidel's conjecture equating symplectic homology with Hochschild homology of a certain Fukaya category.
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| This paper is available on the
arXiv preprint server.
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F. Bourgeois, A. Oancea, The Gysin exact sequence for S^1-equivariant symplectic homology.
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Abstract :
We define S1-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space. We define a parametrized version of the Robbin-Salamon index, which gives the grading for these new versions of symplectic homology. We indicate several applications and ramifications of our constructions.
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| This paper is available on the
arXiv preprint server.
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F. Bourgeois, A. Oancea, Fredholm theory and transversality for the parametrized and for the S^1-invariant symplectic action.
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Abstract :
We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the L2-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic S1-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define S1-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn's unique continuation theorem to a class of elliptic integro-differential inequalities of order two.
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| This paper appeared in
Journal of the European Mathematical Society.
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F. Bourgeois, O. van Koert, Contact homology of left-handed stabilizations and plumbing of open books.
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Abstract :
We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold, we exhibit a closed Reeb orbit that bounds a single finite energy plane like in the computation for the overtwisted case. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. This result can be extended to other contact manifolds by using connected sums. The latter is related to the plumbing or 2-Murasugi sum of contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation.
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| This paper appeared in
Communications in Contemporary Mathematics.
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F. Bourgeois, K. Niederkrüger, Towards a good definition of algebraically overtwisted.
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Abstract :
Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We show that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish algebraically overtwisted.
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| This paper appeared in
Expositiones Mathematicae.
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F. Bourgeois, A survey of contact homology.
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| This paper appeared in a volume of the
CRM Proceedings & Lecture Notes book series.
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F. Bourgeois, A. Oancea, An exact sequence for contact- and symplectic homology.
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Abstract :
A symplectic manifold W with contact type boundary M=∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M). We establish a Gysin-type exact sequence in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to HC(M), by a map of degree -2, which then maps to SH(W). Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M.
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| This paper appeared in
Inventiones Mathematicae.
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F. Bourgeois, A. Oancea, Symplectic Homology, autonomous Hamiltonians, and Morse-Bott moduli spaces.
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Abstract :
We define Floer homology for a time-independent or autonomous Hamiltonian on
a symplectic manifold with contact-type boundary under the assumption that its 1-
periodic orbits are transversally nondegenerate. Our construction is based on Morse-
Bott techniques for Floer trajectories. Our main motivation is to understand the
relationship between the linearized contact homology of a fillable contact manifold
and the symplectic homology of its filling.
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| This paper appeared in
Duke Mathematical Journal.
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F. Bourgeois, K. Cieliebak, T. Ekholm, A note on Reeb dynamics on the tight 3-sphere.
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Abstract :
We show that a nondegenerate tight contact form on the 3-sphere
has exactly two simple closed Reeb orbits if and only if the differential
in linearized contact homology vanishes. Moreover, in this case the Floquet
multipliers and Conley-Zehnder indices of the two Reeb orbits agree with
those of a suitable irrational ellipsoid in 4-space.
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| This paper appeared in
Journal of Modern Dynamics.
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F. Bourgeois, Contact homology and homotopy groups of the
space of contact structures.
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Abstract :
Using contact homology, we reobtain some recent results of Geiges and Gonzalo
about the first fundamental group of the space of contact structures on some
3-manifolds. We show that our techniques can be used to study higher
dimensional contact manifolds and higher order homotopy groups.
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| This paper appeared in
Mathematical Research Letters.
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F. Bourgeois, V. Colin, Homologie de contact des
variétés toroïdales.
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Abstract :
We show that contact homology distinguishes infinitely many tight contact
structures on any orientable, toroidal, irreducible 3-manifold. As a consequence
of the contact homology computations, on a very large class of toroidal
manifolds, all known examples of universally tight contact structures
with nonvanishing torsion satisfy the Weinstein conjecture.
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| This paper appeared in
Geometry and Topology.
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F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder,
Compactness results in Symplectic Field Theory.
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Abstract :
This is one in a series of papers devoted to the foundations of Symplectic
Field Theory. We prove compactness results for moduli spaces of holomorphic
curves arising in Symplectic Field Theory. The theorems generalize Gromov's
compactness theorem as well as compactness theorems in Floer homology theory,
and in contact geometry.
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| This paper appeared in
Geometry and Topology.
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F. Bourgeois,
A Morse-Bott approach to contact homology (Ph.D. Thesis).
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Abstract :
Contact homology was introduced by Eliashberg, Givental
and Hofer. In this theory, we count holomorphic curves in the symplectization
of a contact manifold, which are asymptotic to periodic Reeb orbits. These
closed orbits are assumed to be nondegenerate and, in particular, isolated.
This assumption makes practical computations of contact homology very
difficult.
In this thesis, we develop computational methods for contact homology in
Morse-Bott situations, in which closed Reeb orbits form submanifolds of the
contact manifold. We require some Morse-Bott type assumptions on the contact
form, a positivity property for the Maslov index, mild requirements on the Reeb
flow, and c1(ξ) = 0.
We then use these methods to compute contact homology for several examples, in
order to illustrate their efficiency. As an application of these contact
invariants, we show that T5 and T2 × S3
carry infinitely many
pairwise non-isomorphic contact structures in the trivial formal homotopy
class.
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| You can dowload the full text of my Ph.D. thesis in
PostScript or in
PDF format.
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F. Bourgeois,
Odd dimensional tori are contact manifolds.
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Abstract :
We show that, for every contact manifold M and for every Riemann surface
Σ of genus at least 1, the manifold Σ × M admits a contact
structure. In particular, the tori T2n + 1 are contact manifolds.
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| This paper appeared in
International Mathematics Research Notices.
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F. Bourgeois, K. Mohnke,
Coherent Orientations in Symplectic Field Theory.
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Abstract :
We study the coherent orientations of the moduli spaces of 'trajectories' in
Symplectic Field Theory, following the lines of Floer & Hofer. In particular
we examine their behavior at multiple closed Reeb orbits under change of the
asymptotic direction. Analogous to the orientation of the unstable tangent
spaces of critical points in finite dimensional Morse theory, the orientations
are determined by a certain choice of orientation at each closed Reeb orbit.
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| This paper appeared in
Mathematische Zeitschrift.
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F. Bourgeois,
A Morse-Bott approach to Contact Homology.
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Abstract :
Contact homology was introduced by Eliashberg, Givental and Hofer;
this contact invariant is based on J-holomorphic curves in the
symplectization of a contact manifold.
We expose an extension of contact homology to Morse-Bott situations,
in which closed Reeb orbits form submanifolds of the contact manifold.
We then illustrate how to use this to compute contact homology with
several examples. |
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This is a paper announcing the results of my PhD thesis.
You can download it in
PostScript or in
PDF format. |