3 In memoriam

V. I. Arnol'd, June 12, 1937 - June 3, 2010.

The very sad news of his death is reported today here.

After Floer, the main difficulty in solving the weak Arnol'd conjecture on a compact symplectic manifold $M$ lies in defining a Floer chain complex generated by 1-periodic orbits of an arbitrary non-degenerate Hamiltonian $H: S^1\times M \to \mathbb{R}$, in such a way that the homology is independent of $H$. Once one has that, the remaining step (proving an isomorphism with Morse homology) can be done either by a computation with small autonomous Hamiltonians, or by a "PSS" isomorphism.

When $M$ is monotone, the crucial compactness theorems for solutions to Floer's equation (used to define the candidate-differential on the Floer complex, to prove that it squares to zero, and, in a variant, to prove the invariance of the theory) can be proved using index considerations. When $M$ is Calabi-Yau, compactness needs an additional idea, that holomorphic spheres generically don't hit cylinders solving Floer's equation. This is beautifully worked out in

Hofer, H.; Salamon, D. A. "Floer homology and Novikov rings." The Floer memorial volume, 483--524, Progr. Math., 133, Birkhäuser, Basel, 1995; MR1362838.

In general, where there may be holomorphic spheres with small negative Chern number, one has little choice but to allow "stable trajectories" consisting of broken Floer trajectories with holomorphic bubble-trees attached. Transversality is proved by introducing multi-valued perturbations to the equations, and this forces one to use rational coefficients. References:

Fukaya, Kenji; Ono, Kaoru. "Arnold conjecture and Gromov-Witten invariant". Topology 38 (1999), no. 5, 933-1048. MR1688434
Liu, Gang; Tian, Gang, "Floer homology and Arnold conjecture", J. Differential Geom. 49 (1998), no. 1, 1-74. MR1642105

[Edit: both these references offer proofs of the weak Arnol'd conjecture with rational coefficients.] For a detailed introduction to these "virtual transversality" methods, see

Salamon, Dietmar, "Lectures on Floer homology". MR1702944

The technical complications of virtual transversality theory are notorious, and one could wish for a fully detailed textbook account.

What's left?

So far as I know, there is no proof for general manifolds that the number $h$ of 1-periodic orbits of a non-degenerate Hamiltonian is at least the sum of the mod $p$ Betti numbers. The strong Arnol'd conjecture for non-degenerate Hamiltonians, that $h$ is at least the minimum number of critical points of a Morse function, is wide open.

2 Minor clean-up

The

After Floer, the main problem difficulty in solving the weak Arnol'd conjecture on a compact symplectic manifold $M$ lies in defining a Floer chain complex generated by 1-periodic orbits of an arbitrary non-degenerate Hamiltonian $H: S^1\times M \to \mathbb{R}$, in such a way that the homology is independent of $H$. Once one has that, the remaining step (proving an isomorphism with Morse homology) can be done either by a computation with small autonomous Hamiltonians, or by a "PSS" isomorphism.

When $M$ is monotone, the crucial compactness theorems for solutions to Floer's equation (used to define the candidate-differential on the Floer complex, to prove that it squares to zero, and, in a variant, to prove the invariance of the theory) can be proved using index considerations. When $M$ is Calabi-Yau, compactness needs an additional idea, that holomorphic spheres generically don't hit cylinders solving Floer's equation. This is beautifully worked out in

Hofer, H.; Salamon, D. A. "Floer homology and Novikov rings." The Floer memorial volume, 483--524, Progr. Math., 133, Birkhäuser, Basel, 1995; MR1362838.

In general, where there may be holomorphic spheres with small negative Chern number, one has little choice but to allow "stable trajectories" consisting of broken Floer trajectories with holomorphic bubble-trees attached. Transversality is proved by introducing multi-valued perturbations to the equations, and this forces one to use rational coefficients. References:

Fukaya, Kenji; Ono, Kaoru. "Arnold conjecture and Gromov-Witten invariant". Topology 38 (1999), no. 5, 933-1048. MR1688434
Liu, Gang; Tian, Gang, "Floer homology and Arnold conjecture", J. Differential Geom. 49 (1998), no. 1, 1-74. MR1642105

[Edit: both these references offer proofs of the weak Arnol'd conjecture with rational coefficients.] For a detailed introduction to the these "virtual transversality" methods, see

Salamon, Dietmar, "Lectures on Floer homology". MR1702944

The technical complications of virtual transversality theory are notorious, and one could wish for a fully detailed textbook account.

What's left?

So far as I know, there is no proof for general manifolds that the number $h$ of Hamiltonian 1-periodic orbits of a non-degenerate Hamiltonian is at least the sum of the mod $p$ Betti numbers. The "strong" strong Arnol'd conjecture for non-degenerate Hamiltonians, that $h$ is at least the minimum number of critical points of a Morse function, remains is wide open.

1

The main problem in solving the weak Arnol'd conjecture on a compact symplectic manifold $M$ lies in defining a Floer chain complex generated by 1-periodic orbits of an arbitrary non-degenerate Hamiltonian $H: S^1\times M \to \mathbb{R}$, in such a way that the homology is independent of $H$. Once one has that, the remaining step (proving an isomorphism with Morse homology) can be done either by a computation with small autonomous Hamiltonians, or by a "PSS" isomorphism.

When $M$ is monotone, the crucial compactness theorems for solutions to Floer's equation (used to define the candidate-differential on the Floer complex, to prove that it squares to zero, and, in a variant, to prove the invariance of the theory) can be proved using index considerations. When $M$ is Calabi-Yau, compactness needs an additional idea, that holomorphic spheres generically don't hit cylinders solving Floer's equation. This is beautifully worked out in

Hofer, H.; Salamon, D. A. "Floer homology and Novikov rings." The Floer memorial volume, 483--524, Progr. Math., 133, Birkhäuser, Basel, 1995; MR1362838.

In general, where there may be holomorphic spheres with small negative Chern number, one has little choice but to allow "stable trajectories" consisting of broken Floer trajectories with holomorphic bubble-trees attached. Transversality is proved by introducing multi-valued perturbations to the equations, and this forces one to use rational coefficients. References:

Fukaya, Kenji; Ono, Kaoru. "Arnold conjecture and Gromov-Witten invariant". Topology 38 (1999), no. 5, 933-1048. MR1688434
Liu, Gang; Tian, Gang, "Floer homology and Arnold conjecture", J. Differential Geom. 49 (1998), no. 1, 1-74. MR1642105

For a detailed introduction to the virtual methods, see

Salamon, Dietmar, "Lectures on Floer homology". MR1702944

The technical complications of virtual transversality theory are notorious, and one could wish for a fully detailed textbook account.

What's left?

So far as I know, there is no proof for general manifolds that the number $h$ of Hamiltonian orbits of a non-degenerate Hamiltonian is at least the sum of the mod $p$ Betti numbers. The "strong" Arnol'd conjecture, that $h$ is at least the minimum number of critical points of a Morse function, remains wide open.