Symplectic fixed points and holomorphic spheres

@article{Floer1989SymplecticFP,
  title={Symplectic fixed points and holomorphic spheres},
  author={Andreas Floer},
  journal={Communications in Mathematical Physics},
  year={1989},
  volume={120},
  pages={575-611}
}
  • A. Floer
  • Published 1 December 1989
  • Mathematics
  • Communications in Mathematical Physics
LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we… 
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References

SHOWING 1-10 OF 43 REFERENCES
A relative morse index for the symplectic action
The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define
Pseudo holomorphic curves in symplectic manifolds
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called
A symplectic fixed point theorem for ℂℙn
SummaryTwo symplectic diffeomorphisms,φ0,φ1 of a symplectic manifold (X, ω) are said to be homologous if there exists a smooth homotopyφ1,t∋[0, 1] of symplectic diffeomorphisms between them such that
The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold
Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of mass
A refinement of the Conley index and an application to the stability of hyperbolic invariant sets
  • A. Floer
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1987
Abstract A compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation
A Morse equation in Conley's index theory for semiflows on metric spaces
Abstract Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates
Examples of symplectic structures
SummaryIn this paper we construct symplectic forms $$\tilde \omega _k , k \geqq 0$$ , on a compact manifold $${\tilde Y}$$ which have the same homotopy theoretic invariants, but which are not
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