Alexandru Oancea
On the topology of fillings of contact manifolds and applications.
Wed, 12/02/2009 - 16:01 — SomNambulAuthors: Alexandru Oancea, Claude Viterbo
Subjects: Symplectic Geometry
AbstractThe aim of this paper is to address the following question: given a contact
manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic
manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem
of Eliashberg, Floer and McDuff to prove that under suitable assumptions the
map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We
then apply this method in the case of contact manifolds having a contact
embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold.The Gysin exact sequence for $S^1$-equivariant symplectic homology.
Sat, 09/26/2009 - 04:10 — SomNambulAuthors: Frédéric Bourgeois, Alexandru Oancea
Subjects: Symplectic Geometry
AbstractWe define $S^1$-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.Fredholm theory and transversality for the parametrized and $S^1$-invariant symplectic action.
Tue, 09/15/2009 - 11:27 — SomNambulAuthors: Frédéric Bourgeois, Alexandru Oancea
Subjects: Symplectic Geometry
AbstractWe study the parametrized Hamiltonian action functional for
finite-dimensional families of Hamiltonians. We show that the linearized
operator for the $L^2$-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 $S^1$-invariant families of
Hamiltonians and almost complex structures, parametrized by odd-dimensional
spheres. This is a foundational result used to define $S^1$-equivariant Floer
homology.Fredholm theory and transversality for the parametrized and $S^1$-invariant symplectic action.
Tue, 09/15/2009 - 11:27 — SomNambulAuthors: Frédéric Bourgeois, Alexandru Oancea
Subjects: Symplectic Geometry
AbstractWe study the parametrized Hamiltonian action functional for
finite-dimensional families of Hamiltonians. We show that the linearized
operator for the $L^2$-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 $S^1$-invariant families of
Hamiltonians and almost complex structures, parametrized by odd-dimensional
spheres. This is a foundational result used to define $S^1$-equivariant Floer
homology.
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