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Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. (English) Zbl 1158.53067

Let \(H\) be a Hamiltonian on a symplectic manifold \((W,\omega)\). One of the most important assumptions in the Floer homology is that the \(1\)-periodic orbits of the Hamiltonian vector field \(X_ H\) are non-degenerate, which makes \(H\) time-dependent.
In this paper, the authors generalize the notion of Floer homology by defining it for a time-independent or autonomous Hamiltonian \(H: W\to \mathbb R\) under the assumption that its \(1\)-periodic orbits are transversally non-degenerate. There is a natural class of time-independent Hamiltonians on \(W\) whose non-constant \(1\)-periodic orbits correspond precisely to closed Reeb orbits on \(M= \partial W\) and for which the Floer trajectories can be related to holomorphic cylinders in the symplectization \(M\times\mathbb R\). The authors relate the Floer trajectories of a specific time-dependent perturbation to the Floer trajectories of the unperturbed Hamiltonian in the way that Floer homology for time-independent Hamiltonians serves as a bridge between symplectic homology and linearized contact homology.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
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[1] F. Bourgeois, A Morse-Bott approach to contact homology , Ph.D. dissertation, Stanford University, Stanford, Calif., 2002. · Zbl 1046.57017
[2] F. Bourgeois and K. Mohnke, Coherent orientations in symplectic field theory , Math. Z. 248 (2004), 123–146. · Zbl 1060.53080
[3] F. Bourgeois and A. Oancea, An exact sequence for contact- and symplectic homology , to appear in Invent. Math., preprint,\arxiv0704.2169v2[math.SG] · Zbl 1167.53071
[4] -, The Gysin exact sequence for \(S^1\)-equivariant symplectic homology , in preparation. · Zbl 1405.53121
[5] K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture , J. Eur. Math. Soc. (JEMS) 4 (2002), 115–142. · Zbl 1012.53066
[6] K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology, II: A general construction , Math. Z. 218 (1995), 103–122. · Zbl 0869.58011
[7] K. Cieliebak, A. Floer, H. Hofer, and K. Wysocki, Applications of symplectic homology, II: Stability of the action spectrum , Math. Z. 223 (1996), 27–45. · Zbl 0869.58013
[8] A. Floer, Symplectic fixed points and holomorphic spheres , Comm. Math. Phys. 120 (1989), 575–611. · Zbl 0755.58022
[9] A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry , Math. Z. 212 (1993), 13–38. · Zbl 0789.58022
[10] -, Symplectic homology, I: Open sets in \(\C^n\) , Math. Z. 215 (1994), 37–88. · Zbl 0810.58013
[11] A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic action , Duke Math. J. 80 (1995), 251–292. · Zbl 0846.58025
[12] U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology , Int. Math. Res. Not. 2004 , no. 42, 2179–2269. · Zbl 1088.53058
[13] H. Hofer and D. A. Salamon, “Floer homology and Novikov rings” in The Floer Memorial Volume , Progr. Math. 133 , Birkhäuser, Basel, 1995, 483–524. · Zbl 0842.58029
[14] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations, I: Asymptotics , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337–379. · Zbl 0861.58018
[15] -, “Properties of pseudoholomorphic curves in symplectisations, IV: Asymptotics with degeneracies” in Contact and Symplectic Geometry (Cambridge, 1994) , Publ. Newton Inst. 8 , Cambridge Univ. Press, Cambridge, 1996, 78–117. · Zbl 0868.53043
[16] D. Mcduff and D. Salamon, \(J\)-Holomorphic Curves and Symplectic Topology , Amer. Math. Soc. Colloq. Publ. 52 , Amer. Math. Soc. Providence, 2004. · Zbl 1064.53051
[17] A. Oancea, “A survey of Floer homology for manifolds with contact type boundary or symplectic homology” in Symplectic Geometry and Floer Homology: A Survey of Floer Homology for Manifolds with Contact Type Boundary or Symplectic Homology , Ensaios Mat. 7 , Soc. Brasil. Mat., Rio de Janeiro, 2004, 51–91. · Zbl 1070.53056
[18] J. Robbin and D. Salamon, The Maslov index for paths , Topology 32 (1993), 827–844. · Zbl 0798.58018
[19] D. Salamon, “Lectures on Floer homology” in Symplectic Geometry and Topology (Park City, Utah, 1997) , IAS/Park City Math. Ser. 7 , Amer. Math. Soc., Providence, 1999, 143–229. · Zbl 1031.53118
[20] D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index , Comm. Pure Appl. Math. 45 (1992), 1303–1360. · Zbl 0766.58023
[21] M. Schwarz, Cohomology operations from \(S^1\)-cobordisms in Floer homology , Ph.D. dissertation, Eidgenössische Technische Hochschule Zürich, Zürich, 1995, no. 11182.
[22] I. Ustilovsky, Contact homology and contact structures on \(S^4m+1\) , Ph.D. dissertation, Stanford University, Stanford, Calif., 1999. · Zbl 1034.53080
[23] C. Viterbo, Functors and computations in Floer homology with applications, I , Geom. Funct. Anal. 9 (1999), 985–1033. · Zbl 0954.57015
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