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Three approaches towards Floer homology of cotangent bundles  [PDF]
Joa Weber
Mathematics , 2006, DOI: 10.4310/JSG.2005.v3.n4.a7
Abstract: Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and Abbondandolo-Schwarz (2004). The theory is illustrated by calculating Morse and Floer homology in case of the euclidean n-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.
Corrigendum: "On the Floer homology of cotangent bundles"  [PDF]
Alberto Abbondandolo,Matthias Schwarz
Mathematics , 2013,
Abstract: We fix an orientation issue which appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel-Whitney class of the underlying manifold does not vanish on 2-tori, this isomorphism requires the use of a twisted version of the Floer complex.
On the Floer homology of cotangent bundles  [PDF]
Alberto Abbondandolo,Matthias Schwarz
Mathematics , 2004, DOI: 10.1002/cpa.2009
Abstract: This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle of a compact orientable manifold M. The first result is a new uniform estimate for the solutions of the Floer equation, which allows to deal with a larger - and more natural - class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M, in the periodic case, or of the based loop space of M, in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian which is the Legendre transform of a Lagrangian on TM, and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of free or based loops on M of Sobolev class W(1,2).
Lagrangian Rabinowitz Floer homology and twisted cotangent bundles  [PDF]
Will J. Merry
Mathematics , 2010,
Abstract: We study the following rigidity problem in symplectic geometry:can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser. Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the `virtually contact' setting. By means of an Abbondandolo-Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.
Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems  [PDF]
Kei Irie
Mathematics , 2010,
Abstract: The objective of this note is to prove an existence result for brake orbits in classical Hamiltonian systems (which was first proved by S.V.Bolotin) by using Floer theory. To this end, we compute an open string analogue of symplectic homology (so called wrapped Floer homology) of some domains in cotangent bundles, which appear naturally in the study of classical Hamiltonian systems. The main part of the computations is to show invariance of wrapped Floer homology under certain handle attaching to domains.
Floer homology of cotangent bundles and the loop product  [PDF]
Alberto Abbondandolo,Matthias Schwarz
Mathematics , 2008,
Abstract: We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.
On the Rabinowitz Floer homology of twisted cotangent bundles  [PDF]
Will J. Merry
Mathematics , 2010,
Abstract: Consider the cotangent bundle of a Riemannian manifold $(M,g)$ of dimension 2 or more, endowed with a twisted symplectic structure defined by a closed weakly exact 2-form $\sigma$ on $M$ whose lift to the universal cover of $M$ admits a bounded primitive. We compute the Rabinowitz Floer homology of energy hypersurfaces $\Sigma_{k}=H^{-1}(k)$ of mechanical (kinetic energy + potential) Hamiltonians $H$ for the case when the energy value k is greater than the Mane critical value c. Under the stronger condition that k>c_{0}, where c_{0} denotes the strict Mane critical value, Abbondandolo and Schwarz recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k>c, thus covering cases where $\sigma$ is not exact. As a consequence, we deduce that the hypersurface corresponding to the energy level k is never displaceable for any k>c. Moreover, we prove that if dim M > 1, the homology of the free loop space of $M$ is infinite dimensional, and if the metric is chosen generically, a generic Hamiltonian diffeomorphism has infinitely many leaf-wise intersection points in $\Sigma_{k}$.
Propagation in Hamiltonian dynamics and relative symplectic homology  [PDF]
Paul Biran,Leonid Polterovich,Dietmar Salamon
Mathematics , 2001,
Abstract: The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.
Hofer-Zehnder capacity of unit disk cotangent bundles and the loop product  [PDF]
Kei Irie
Mathematics , 2011,
Abstract: We prove the finiteness of the Hofer-Zehnder capacity of unit disk cotangent bundles of closed Riemannian manifolds, under some simple topological assumptions on the manifolds. The key ingredient of the proof is a computation of the pair-of-pants product on Floer homology of cotangent bundles. We reduce it to a simple computation of the loop product, making use of results of A.Abbondandolo- M.Schwarz.
Floer homology and the heat flow  [PDF]
Dietmar A. Salamon,Joa Weber
Mathematics , 2003, DOI: 10.1007/s00039-006-0577-4
Abstract: We study the heat flow in the loop space of a closed Riemannian manifold $M$ as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space.
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