## Duke Mathematical Journal

### Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces

#### Abstract

We define Floer homology for a time-independent or autonomous Hamiltonian on a symplectic manifold with contact-type boundary under the assumption that its $1$-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between the linearized contact homology of a fillable contact manifold and the symplectic homology of its filling

#### Article information

Source
Duke Math. J. Volume 146, Number 1 (2009), 71-174.

Dates
First available in Project Euclid: 17 December 2008

http://projecteuclid.org/euclid.dmj/1229530285

Digital Object Identifier
doi:10.1215/00127094-2008-062

Zentralblatt MATH identifier
1158.53067

Mathematical Reviews number (MathSciNet)
MR2475400

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects

#### Citation

Bourgeois, Frédéric; Oancea, Alexandru. Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. Duke Mathematical Journal 146 (2009), no. 1, 71--174. doi:10.1215/00127094-2008-062. http://projecteuclid.org/euclid.dmj/1229530285.

#### References

• F. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. dissertation, Stanford University, Stanford, Calif., 2002.
• F. Bourgeois and K. Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004), 123--146.
• F. Bourgeois and A. Oancea, An exact sequence for contact- and symplectic homology, to appear in Invent. Math., preprint,\arxiv0704.2169v2[math.SG]
• —, The Gysin exact sequence for $S^1$-equivariant symplectic homology, in preparation.
• K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. (JEMS) 4 (2002), 115--142.
• K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology, II: A general construction, Math. Z. 218 (1995), 103--122.
• 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.
• A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575--611.
• A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993), 13--38.
• —, Symplectic homology, I: Open sets in $\C^n$, Math. Z. 215 (1994), 37--88.
• A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), 251--292.
• U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. 2004, no. 42, 2179--2269.
• H. Hofer and D. A. Salamon, Floer homology and Novikov rings'' in The Floer Memorial Volume, Progr. Math. 133, Birkhäuser, Basel, 1995, 483--524.
• 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.
• —, 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.
• D. Mcduff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc. Providence, 2004.
• 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.
• J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827--844.
• 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.
• 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.
• M. Schwarz, Cohomology operations from $S^1$-cobordisms in Floer homology, Ph.D. dissertation, Eidgenössische Technische Hochschule Zürich, Zürich, 1995, no. 11182.
• I. Ustilovsky, Contact homology and contact structures on $S^4m+1$, Ph.D. dissertation, Stanford University, Stanford, Calif., 1999.
• C. Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999), 985--1033.