What is the current status of the Arnold conjecture?

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Let $(M, omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth function on $M$.



The conjecture was proved in a weaker (homological) version using Floer homology, with two additional conditions:



  1. Compactness of $M$;

  2. Non-degeneracy of the fixed points of the considered Hamiltonian symplectomorphism.

In this setting, it is now fully understood that the number of fixed points is at least the sum of Betti number of $M$.



I am wondering where the general statement stands today. More precisely:



  1. is compactness really necessary to Floer's framework ?

  2. what is known for degenerate Hamiltonians ?

  3. what is known for the critical points lower bound, rather than the Betti sum ?

Thanks a lot






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    Let $(M, omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth function on $M$.



    The conjecture was proved in a weaker (homological) version using Floer homology, with two additional conditions:



    1. Compactness of $M$;

    2. Non-degeneracy of the fixed points of the considered Hamiltonian symplectomorphism.

    In this setting, it is now fully understood that the number of fixed points is at least the sum of Betti number of $M$.



    I am wondering where the general statement stands today. More precisely:



    1. is compactness really necessary to Floer's framework ?

    2. what is known for degenerate Hamiltonians ?

    3. what is known for the critical points lower bound, rather than the Betti sum ?

    Thanks a lot






    sg.symplectic-geometry symplectic-topology







    share|cite|improve this question























      up vote
      4
      down vote

      favorite









      up vote
      4
      down vote

      favorite











      Let $(M, omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth function on $M$.



      The conjecture was proved in a weaker (homological) version using Floer homology, with two additional conditions:



      1. Compactness of $M$;

      2. Non-degeneracy of the fixed points of the considered Hamiltonian symplectomorphism.

      In this setting, it is now fully understood that the number of fixed points is at least the sum of Betti number of $M$.



      I am wondering where the general statement stands today. More precisely:



      1. is compactness really necessary to Floer's framework ?

      2. what is known for degenerate Hamiltonians ?

      3. what is known for the critical points lower bound, rather than the Betti sum ?

      Thanks a lot






      sg.symplectic-geometry symplectic-topology







      share|cite|improve this question













      Let $(M, omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth function on $M$.



      The conjecture was proved in a weaker (homological) version using Floer homology, with two additional conditions:



      1. Compactness of $M$;

      2. Non-degeneracy of the fixed points of the considered Hamiltonian symplectomorphism.

      In this setting, it is now fully understood that the number of fixed points is at least the sum of Betti number of $M$.



      I am wondering where the general statement stands today. More precisely:



      1. is compactness really necessary to Floer's framework ?

      2. what is known for degenerate Hamiltonians ?

      3. what is known for the critical points lower bound, rather than the Betti sum ?

      Thanks a lot






      sg.symplectic-geometry symplectic-topology




      sg.symplectic-geometry symplectic-topology






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      asked 1 hour ago









      BrianT

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          A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.






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          • Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
            – BrianT
            53 mins ago










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          up vote
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          A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.






          share|cite|improve this answer




















          • Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
            – BrianT
            53 mins ago














          up vote
          4
          down vote













          A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.






          share|cite|improve this answer




















          • Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
            – BrianT
            53 mins ago












          up vote
          4
          down vote










          up vote
          4
          down vote









          A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.






          share|cite|improve this answer












          A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 1 hour ago









          Carlo Beenakker

          69.7k8155261




          69.7k8155261











          • Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
            – BrianT
            53 mins ago
















          • Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
            – BrianT
            53 mins ago















          Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
          – BrianT
          53 mins ago




          Thanks. Is there a reason why we always assume compactness of the symplectic manifold ?
          – BrianT
          53 mins ago

















           

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