Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-14T13:51:52.617Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations

Published online by Cambridge University Press:  05 September 2011

YOUNGJIN BAE  and
URS FRAUENFELDER
YOUNGJIN BAE
Affiliation:
Department of Mathematics and Research Institute of Mathematics, Seoul National University, San 56-1 Sillim-Dong, Gwanak-Gu, Seoul 151-747, Korea. e-mail: jini0919@snu.ac.kr and frauenf@snu.ac.kr
URS FRAUENFELDER
Affiliation:
Department of Mathematics and Research Institute of Mathematics, Seoul National University, San 56-1 Sillim-Dong, Gwanak-Gu, Seoul 151-747, Korea. e-mail: jini0919@snu.ac.kr and frauenf@snu.ac.kr
Get access Rights & Permissions [Opens in a new window]

Abstract

Will 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 471 - 502
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abbondandolo, A. and Schwarz, M.Estimates and computations in Rabinowitz-Floer homology. J. Topology Anal. 1 (2009), no. 4, 307–405.CrossRefGoogle Scholar
[2]Albers, P. and Frauenfelder, U.Spectral invariants in Rabinowitz Floer homology and global Hamiltonian perturbations. J. Mod. Dyn. 4 (2010), no. 2, 329–357.CrossRefGoogle Scholar
[3]Cieliebak, K. and Frauenfelder, U.A Floer homology for exact contact embeddings. Pacific J. Math. 239 (2009), no. 2, 251–316.CrossRefGoogle Scholar
[4]Cieliebak, K. and Frauenfelder, U.Morse homology on noncompact manifolds. J. Korean Math. Soc. 48 (2011), no. 4, 749–774.CrossRefGoogle Scholar
[5]Cieliebak, K., Frauenfelder, U. and Oancea, A.Rabinowitz Floer homology and symplectic homology. Ann. Sci. École Norm. Sup. (4) 43 (2010), no. 6, 957–1015.CrossRefGoogle Scholar
[6]Cieliebak, K., Frauenfelder, U. and Paternain, G.Symplectic topology of Mañé's critical value. Geom. Topol. 14 (2010), 1765–1870.CrossRefGoogle Scholar
[7]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P. Word Processing in Groups (Jones and Bartlett Publishers, 1992)CrossRefGoogle Scholar
[8]Floer, A.Morse theory for Lagrangian intersections. J. Diff. Geom. 28 (1988), 513–547.Google Scholar
[9]Floer, A.The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), 775–813.CrossRefGoogle Scholar
[10]Floer, A.Wittens complex and infinite dimensional Morse theory. J. Diff. Geom. 30 (1989), 207–221.Google Scholar
[11]Frauenfelder, U.The Arnold-Givental conjecture and moment Floer homology. Int. Math. Res. Not. 42 (2004), 2179–2269.CrossRefGoogle Scholar
[12]Gromov, M.Asymptotic invariants in group theory. Geometric group theory II, G.A. Niblo and M.A. Roller, London Math. Soc. Lecture Notes 182 (1993).Google Scholar
[13]Gromov, M.Metric structures for Riemannian and non-Riemannian spaces. Progr. 152 (1999).Google Scholar
[14]Merry, W. On the Rabinowitz Floer homology of twisted cotangent bundles (2010), arXiv:1002.0162, to appear in Calc. Var. Partial Differential Equations.CrossRefGoogle Scholar
[15]Merry, W. and Paternain, G. Index computations in Rabinowitz Floer homology. arXiv:1009.3870, to appear in J. Fixed Point Theory Appl.Google Scholar
[16]Miranda, J.Generic properties for magnetic flows on surfaces. Nonlinearity 19 (2006), 1849–1874.CrossRefGoogle Scholar
[17]Paternain, G.Magnetic rigidity of horocycle flows. Pacific J. Math. 225 (2006), 301–323.CrossRefGoogle Scholar
[18]Piunikhin, S., Salamon, D. and Schwarz, M.Symplectic Floer-Donaldson theory and quantum cohomology. Publ. Newton. Inst. 8, ed. by Thomas, C. B. (Cambridge University Press, 1996), pp 171–200.Google Scholar
[19]Polterovich, L.Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150 (2002) no. 3, 655–686.CrossRefGoogle Scholar
[20]Ritter, A. F.Deformations of symplectic cohomology and exact Lagrangians in ALE spaces. GAFA 20 (2010), no. 3, 779–816.Google Scholar
[21]Robbin, J. and Salamon, D.The Maslov index for paths. Topology 32 (1993), no. 4, 827–884CrossRefGoogle Scholar
[22]Salamon, D. Lectures on Floer homology. In Symplectic Geometry and Topology, eds. Eliashberg, Y. and Traynor, L. IAS/Park City Math. Series, vol. 7, AMS (1999), 143–229.Google Scholar
[23]Schwarz, M. Morse homology. Prog. Math. vol. 111 (1993).CrossRefGoogle Scholar
[24]Sikorav, J.-C.Growth of a primitive of a differential form. Bull. Soc. Math. France. 129 2, (2001), 159–168.CrossRefGoogle Scholar
[25]Viterbo, C.Functors and computations in Floer homology with applications I. Geom. Funct. Anal. 9 (1998), 985–1033.CrossRefGoogle Scholar