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Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action

Frédéric Bourgeois, Alexandru Oancea (2010)

Journal of the European Mathematical Society

We 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...

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

We show that if ( M , ω ) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of  ( M , ω ) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in  M × M when M satisfies...

Hofer–Zehnder capacity of unit disk cotangent bundles and the loop product

Kei Irie (2014)

Journal of the European Mathematical Society

We prove a new finiteness result for the Hofer–Zehnder capacity of certain unit disk cotangent bundles. It is proved by a computation of the pair-of-pants product on Floer homology of cotangent bundles, combined with the theory of spectral invariants. The computation of the pair-of-pants product is reduced to a simple key computation of the Chas–Sullivan loop product.

Inner and outer hamiltonian capacities

David Hermann (2004)

Bulletin de la Société Mathématique de France

The aim of this paper is to compare two symplectic capacities in n related with periodic orbits of Hamiltonian systems: the Floer-Hofer capacity arising from symplectic homology, and the Viterbo capacity based on generating functions. It is shown here that the inner Floer-Hofer capacity is not larger than the Viterbo capacity and that they are equal for open sets with restricted contact type boundary. As an application, we prove that the Viterbo capacity of any compact Lagrangian submanifold is...

Length minimizing Hamiltonian paths for symplectically aspherical manifolds

Ely Kerman, François Lalonde (2003)

Annales de l’institut Fourier

In this note we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of Polterovich and Schwarz, we study the role, in the Floer complex of the generating Hamiltonian, of the global extrema which remain fixed as the time varies. Our main result determines a natural condition which implies that the corresponding path minimizes the positive Hofer length. We use this to prove that a quasi-autonomous Hamiltonian...

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

For a Morse function f on a compact oriented manifold M , we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of f in terms of the Betti numbers of M and the behavior of f with respect to links....

Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Robert Lipshitz, David Treumann (2016)

Journal of the European Mathematical Society

Let A be a dg algebra over 𝔽 2 and let M be a dg A -bimodule. We show that under certain technical hypotheses on A , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product M A L M and converges to the Hochschild homology of M . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

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