Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology

Fukaya, Kenji

doi:10.1007/1-4020-4266-3_06

Cited by 51 publications

(63 citation statements)

References 24 publications

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“…For instance, it leads to a quick proof of Fukaya's recent result in [6], which we may state in the following general form: if S 1 ×S n−1 admits a Lagrangian embedding in a symplectic manifold in such a way that it is displaceable by a Hamiltonian isotopy, then {2, n} ∩ Im(µ) = ∅ for n even and {2, 3 − n} ∩ Im(µ) = ∅ for n odd. This map induces a morphism in homology whose non-triviality is used to detect the existence of periodic orbits of Hamiltonian diffeomorphisms that separate L from L .…”

Section: Constraints On Maslov Indicesmentioning

confidence: 98%

Cluster homology: An overview of the construction and results

Cornea¹,

Lalonde²

2006

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We associate, to a Lagrangian submanifold L of a symplectic manifold, a new homology, called the cluster homology of L, which is invariant up to ambient symplectic diffeomorphisms. We discuss various applications concerning analytical, topological, and dynamical properties of Lagrangian submanifolds. We also deduce a new universal Floer homology, defined without obstruction, for pairs of Lagrangian submanifolds.

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Section: Constraints On Maslov Indicesmentioning

confidence: 98%

Cluster homology: An overview of the construction and results

Cornea¹,

Lalonde²

2006

Electron. Res. Announc. Amer. Math. Soc.

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“…Because of the C n factor, there is a compactly supported Hamiltonian isotopy which displaces L from itself. Following a strategy which goes back to Gromov's introduction of pseudo-holomorphic curves to symplectic topology in [13], and whose applications to Lagrangian embeddings have been developed among others by Polterovich, Oh, Biran-Cieliebak, and Fukaya [22], [21], [5], [11], the choice of such a displacing isotopy determines a one parameter family deformation of the Cauchy-Riemann equation. Under generic conditions, the corresponding moduli space of solutions in the constant homotopy class becomes a manifold of dimension 2n with boundary diffeomorphic to L.…”

Section: Introductionmentioning

confidence: 99%

Framed bordism and Lagrangian embeddings of exotic spheres

Abouzaid

2012

Ann. Math.

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In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of the standard sphere. The main ingredients of the construction are (1) the fact that the graph of the Hopf fibration embeds the standard sphere, and hence any Lagrangian which embeds in its cotangent bundle, as a displaceable Lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space, and (2) a moduli space of solutions to a perturbed CauchyRiemann equation introduced by Gromov.

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“…In this way, Fukaya combined Gromov and Polterovich's pseudo-holomorphic curve approach and Viterbo's loop space approach [V1], and proved several new results on the structure of Lagrangian embeddings in C n . The following are some sample results proven by this method [Fu4] :…”

Section: Conjecture 42 (Maslov Class Conjecture) Any Compact Lagranmentioning

confidence: 99%

“…Recently Fukaya [Fu4] gave a new construction of the A ∞ -structure described in the previous section as a deformation of the differential graded algebra of de Rham complex of L associated to a natural solution to the Maurer-Cartan equation of the Batalin-Vilkovisky structure discovered by Chas and Sullivan [CS] on the loop space. In this way, Fukaya combined Gromov and Polterovich's pseudo-holomorphic curve approach and Viterbo's loop space approach [V1], and proved several new results on the structure of Lagrangian embeddings in C n .…”

Section: Conjecture 42 (Maslov Class Conjecture) Any Compact Lagranmentioning

confidence: 99%

Floer homology in symplectic geometry and in mirror symmetry

Oh¹,

Fukaya²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. In this article, the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory leads to the C 0 symplectic invariants of Hamiltonian flows and to the study of topological Hamiltonian dynamics. In the second case, the authors explain how Floer's original construction of Lagrangian intersection Floer homology is obstructed in general as soon as one leaves the category of exact Lagrangian submanifolds. They will survey construction, obstruction and promotion of the Floer complex to the A∞ category of symplectic manifolds. Some applications of this general machinery to the study of the topology of Lagrangian embeddings in relation to symplectic topology and to mirror symmetry are also reviewed. PrologueThe Darboux theorem in symplectic geometry manifests flexibility of the group of symplectic transformations. On the other hand, the following celebrated theorem of Eliashberg [El1] revealed subtle rigidity of symplectic transformations : The subgroup Symp(M, ω) consisting of symplectomorphisms is closed in Dif f (M ) with respect to the C 0 -topology. This demonstrates that the study of symplectic topology is subtle and interesting. Eliashberg's theorem relies on a version of non-squeezing theorem as proven by Gromov [Gr]. Gromov [Gr] uses the machinery of pseudo-holomorphic curves to prove his theorem. There is also a different proof by Ekeland and Hofer [EH] of the classical variational approach to Hamiltonian systems. The interplay between these two facets of symplectic geometry has been the main locomotive in the development of symplectic topology since Floer's pioneering work on his 'semi-infinite' dimensional homology theory, now called the Floer homology theory.As in classical mechanics, there are two most important boundary conditions in relation to Hamilton's equationẋ = X H (t, x) on a general symplectic manifold : one is the periodic boundary condition γ(0) = γ(1), and the other is the LagrangianThe latter replaces the two-point boundary condition in classical mechanics.Key words and phrases. Floer homology, Hamiltonian flows, Lagrangian submanifolds, A∞-structure, mirror symmetry.Oh thanks A. Weinstein and late A. Floer for putting everlasting marks on his mathematics. Both authors thank H. Ohta and K. Ono for a fruitful collaboration on the Lagrangian intersection Floer theory which some part of this survey is based on.

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Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology

Cited by 51 publications

References 24 publications

Cluster homology: An overview of the construction and results

Cluster homology: An overview of the construction and results

Framed bordism and Lagrangian embeddings of exotic spheres

Floer homology in symplectic geometry and in mirror symmetry

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