Functors and Computations in Floer Homology with Applications, I

Viterbo, Claude

doi:10.1007/s000390050106

Cited by 306 publications

(615 citation statements)

References 43 publications

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“…This is a special case of a generalization of Lemma 4.3 stated as Proposition 1.1 in [43]. We give an ad hoc construction in the case at hand.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Positive topological entropy of Reeb flows on spherizations

Macarini¹,

Schlenk²

2011

Math. Proc. Camb. Phil. Soc.

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Abstract. Let M be a closed manifold whose based loop space Ω(M ) is "complicated". Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T * M which is fiberwise starshaped with respect to the origin. Choose a function H : T * M → Ê such that Σ is a regular energy surface of H, and let ϕ t be the restriction to Σ of the Hamiltonian flow of H. Theorem 1. The topological entropy of ϕ t is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain-Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T * M is positive.The following corollary extends results of Morse and Gromov on the number of geodesics between two points. Corollary 1. Given q ∈ M , for almost every q ′ ∈ M the number of orbits of the flow ϕ t from Σ q to Σ q ′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed orbits.Corollary 2. Let M be a closed surface different from S 2 , ÊP 2 , the torus and the Klein bottle. Then ϕ t carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

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“…This is a special case of a generalization of Lemma 4.3 stated as Proposition 1.1 in [43]. We give an ad hoc construction in the case at hand.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Positive topological entropy of Reeb flows on spherizations

Macarini¹,

Schlenk²

2011

Math. Proc. Camb. Phil. Soc.

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“…We will need the following standard invariance result (cf. [BPS,Poz,Gi9,Vi]): Let us now turn to some examples which are relevant to the proof.…”

Section: Floer Homological Counterpart In the General Casementioning

confidence: 99%

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Ginzburg¹,

Gürel²

2009

Comment. Math. Helv.

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Abstract. In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.

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“…Symplectic (co)homology of a Liouville manifold is a symplectic invariant based on an extension of Hamiltonian Floer (co)homology to non-compact symplectic manifolds. It was introduced by Viterbo [71] in its current form. We recommend [57] for an excellent introduction to symplectic cohomology and the recent manuscript [5] for more.…”

Section: Now Eqn (1) Becomes An Eilenberg-moore Equivalence (Of Dgamentioning

confidence: 99%

Koszul duality patterns in Floer theory

Etgü

Lekili

2017

Geom. Topol.

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We study symplectic invariants of the open symplectic manifolds X Γ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate (DG-)algebra models of the Fukaya category F(X Γ ) of closed exact Lagrangians in X Γ and the wrapped Fukaya category W(X Γ ). When Γ is a Dynkin tree of type A n or D n (and conjecturally also for E 6 , E 7 , E 8 ), we prove that these models for the Fukaya category F(X Γ ) and W(X Γ ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of X Γ for Γ = A n , D n , based on the Legendrian surgery formula of [14]. In the case that Γ is non-Dynkin, we obtain a spectral sequence that converges to symplectic cohomology whose E 2 -page is given by the Hochschild cohomology of the preprojective algebra associated to the corresponding Γ. It is conjectured that this spectral sequence is degenerate if the ground field has characteristic zero.

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Functors and Computations in Floer Homology with Applications, I

Cited by 306 publications

References 43 publications

Positive topological entropy of Reeb flows on spherizations

Positive topological entropy of Reeb flows on spherizations

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Koszul duality patterns in Floer theory

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