Note on coisotropic Floer homology and leafwise fixed points

  • Received: 01 November 2019 Revised: 01 September 2020 Published: 11 January 2021
  • Primary: 53D40

  • For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

    Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

    The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

    Citation: Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points[J]. Electronic Research Archive, 2021, 29(4): 2553-2560. doi: 10.3934/era.2021001

    Related Papers:

  • For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

    Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

    The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.



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    [1] P. Albers, A note on local floer homology, arXiv: math/0606600.
    [2] P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, (2008), Art. ID rnm134, 56 pp. doi: 10.1093/imrn/rnm134
    [3] Lagrangian intersections, symplectic energy, and areas of holomorphic curves. Duke Math. J. (1998) 95: 213-226.
    [4] Applications of symplectic homology, II, Stability of the action spectrum. Math. Z. (1996) 223: 27-45.
    [5] Morse theory for Lagrangian intersections. J. Differential Geom. (1988) 28: 513-547.
    [6] The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. (1988) 41: 775-813.
    [7] Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. (1989) 120: 575-611.
    [8] Contact structures on product five-manifolds and fibre sums along circles. Math. Ann. (2010) 348: 195-210.
    [9] Local Floer homology and the action gap. J. Symplectic Geom. (2010) 8: 323-357.
    [10] Fragility and persistence of leafwise intersections. Math. Z. (2015) 280: 989-1004.
    [11] Remarks on $A$-branes, mirror symmetry, and the Fukaya category. J. Geom. Phys. (2003) 48: 84-99.
    [12] C.-M. Marle, Sous-variétés de rang constant d'une variété symplectique, Astérisque, 107–108, Soc. Math. France, Paris (1983), 69–86.
    [13] Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.. Comm. Pure Appl. Math. (1993) 46: 949-993.
    [14] Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.". Comm. Pure Appl. Math. (1995) 48: 1299-1302.
    [15] Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices, (1996), 305–346. doi: 10.1155/S1073792896000219
    [16] Y.-G. Oh, Localization of Floer homology of engulfed topological Hamiltonian loop, Commun. Inf. Syst., 13 (2013), no. 4, 399–443.
    [17] (2015) Symplectic Topology and Floer Homology, Vol. 2, Floer homology and its applications.New Mathematical Monographs, 29, Cambridge University Press.
    [18] M. Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, 119–181, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/trans2/196/08
    [19] A Morse estimate for translated points of contactomorphisms of spheres and projective spaces. Geom. Dedicata (2013) 165: 95-110.
    [20] Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings. J. Symplectic Geom. (2010) 8: 95-118.
    [21] F. Ziltener, A Maslov map for coisotropic submanifolds, leaf-wise fixed points and presymplectic non-embeddings, arXiv: 0911.1460.
    [22] F. Ziltener, Leafwise fixed points for $C^0$-small Hamiltonian flows, Int. Math. Res. Not. IMRN, (2019), 2411–2452. doi: 10.1093/imrn/rnx182
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