10
$\begingroup$

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

$\endgroup$
1
  • 2
    $\begingroup$ As stated the conjecture is true for open manifolds. Any open manifold admits a function without critical points. So you would need to shrink the class of functions you want to compare to, but then you better also restrict the class of Hamiltonians. I don't think there is a consensus what would consist a good class of functions. $\endgroup$ – Thomas Rot Oct 15 '18 at 15:24
8
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ Thanks. Is there a reason why we always assume compactness of the symplectic manifold ? $\endgroup$ – BrianT Oct 15 '18 at 11:02
  • $\begingroup$ my understanding is the assumption is needed to avoid non-contractible 1-periodic orbits. $\endgroup$ – Carlo Beenakker Oct 15 '18 at 13:24
  • $\begingroup$ What do you mean ? How is this related to openness or compactness ? $\endgroup$ – BrianT Oct 16 '18 at 12:59
  • $\begingroup$ @BrianT Compactness of $M$ automatically leads to $C^0$-bounds on the Floer trajectories - this is needed for the moduli spaces to be compact, which is crucial in the setup of Floer homology. And indeed, the $C^0$-bound bit is what usually makes establishing any kind of Floer theory on open manifolds hard. $\endgroup$ – Alessio Pellegrini Feb 15 '19 at 0:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.