scholarly journals Computing the Rabinowitz Floer homology of tentacular hyperboloids

2021 ◽  
Vol 17 (0) ◽  
pp. 353
Author(s):  
Alexander Fauck ◽  
Will J. Merry ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Homology Groups ◽  
Weinstein Conjecture ◽  
Singular Homology ◽  
Rabinowitz Floer Homology ◽  
Compact Case ◽  
Chain Map ◽  
A Chain

<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id="M1">\begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id="M2">\begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id="M4">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id="M5">\begin{document}$ \Sigma_0 $\end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id="M6">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.</p>

Rabinowitz Floer Homology for Tentacular Hamiltonians

2020 ◽  
Author(s):  
Federica Pasquotto ◽  
Robert Vandervorst ◽  
Jagna Wiśniewska
Keyword(s):  
Periodic Orbits ◽  
Moduli Spaces ◽  
Level Sets ◽  
General Framework ◽  
Floer Homology ◽  
Contact Manifolds ◽  
Weinstein Conjecture ◽  
Rabinowitz Floer Homology ◽  
Closed Characteristics ◽  
Definition Of

Abstract This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

scholarly journals ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

2009 ◽  
Vol 01 (04) ◽  
pp. 307-405 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
MATTHIAS SCHWARZ
Keyword(s):  
Maximum Principle ◽  
Energy Level ◽  
Exact Sequence ◽  
Floer Homology ◽  
Energy Functional ◽  
Convex Compact ◽  
Closed Geodesics ◽  
Action Functional ◽  
Uniform Estimates ◽  
Rabinowitz Floer Homology

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

scholarly journals Orderability, contact non-squeezing, and Rabinowitz Floer homology

2018 ◽  
Vol 16 (6) ◽  
pp. 1481-1547
Author(s):  
Peter Albers ◽  
Will J. Merry
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology

scholarly journals Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations

2011 ◽  
Vol 151 (3) ◽  
pp. 471-502 ◽  
Author(s):  
YOUNGJIN BAE ◽  
URS FRAUENFELDER
Keyword(s):  
Exact Sequence ◽  
Isoperimetric Inequality ◽  
Short Exact Sequence ◽  
Loop Space ◽  
Floer Homology ◽  
Critical Value ◽  
Alternative Proof ◽  
Rabinowitz Floer Homology ◽  
Space Homology ◽  
Loop Space Homology

AbstractWill J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

scholarly journals EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

2019 ◽  
Vol 150 (4) ◽  
pp. 1937-1964 ◽  
Author(s):  
Hua-Lin Huang ◽  
Zheyan Wan ◽  
Yu Ye
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Bar Resolution ◽  
Chain Map ◽  
A Chain

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

scholarly journals Translated points and Rabinowitz Floer homology

2013 ◽  
Vol 13 (1) ◽  
pp. 201-214 ◽  
Author(s):  
Peter Albers ◽  
Will J. Merry
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology

scholarly journals FLOER HOMOLOGY FOR SYMPLECTOMORPHISM

2009 ◽  
Vol 11 (06) ◽  
pp. 895-936 ◽  
Author(s):  
HAI-LONG HER
Keyword(s):  
Fixed Points ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Homology Groups ◽  
Symplectic Diffeomorphism ◽  
Novikov Ring ◽  
Compact Symplectic Manifold

Let (M,ω) be a compact symplectic manifold, and ϕ be a symplectic diffeomorphism on M, we define a Floer-type homology FH*(ϕ) which is a generalization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕ up to a Hamiltonian isotopy.

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