Arnold conjecture

Let (M, \omega) be a closed (compact without boundary) symplectic manifold. For any smooth function H: M \to {\mathbb R}, the symplectic form \omega induces a Hamiltonian vector field X_H on M defined by the formula :\omega( X_H, \cdot) = dH. The function H is called a Hamiltonian function. Suppose there is a smooth 1-parameter family of Hamiltonian functions H_t \in C^\infty(M), t \in [0,1]. This family induces a 1-parameter family of Hamiltonian vector fields X_{H_t} on M. The family of vector fields integrates to a 1-parameter family of diffeomorphisms \varphi_t: M \to M. Each individual \varphi_t is a called a Hamiltonian diffeomorphism of M. The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M is greater than or equal to the number of critical points of a smooth function on M. ==Weak Arnold conjecture==

Let (M, \omega) be a closed symplectic manifold. A Hamiltonian diffeomorphism \varphi:M \to M is called nondegenerate if its graph intersects the diagonal of M\times M transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on M, called the Morse number of M. In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field {\mathbb F}, namely \sum_{i=0}^{2n} \dim H_i (M; {\mathbb F}). The weak Arnold conjecture says that :\# \{ \text{fixed points of } \varphi \} \geq \sum_{i=0}^{2n} \dim H_i (M; {\mathbb F}) for \varphi : M \to M a nondegenerate Hamiltonian diffeomorphism. ==Arnold–Givental conjecture==

The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds and L' in terms of the Betti numbers of L, given that L' intersects transversally and L' is Hamiltonian isotopic to . Let (M, \omega) be a compact 2n-dimensional symplectic manifold, let L \subset M be a compact Lagrangian submanifold of M, and let \tau : M \to M be an anti-symplectic involution, that is, a diffeomorphism \tau : M \to M such that \tau^* \omega = -\omega and \tau^2 = \text{id}_M, whose fixed point set is L. Let H_t\in C^\infty(M), t \in [0,1] be a smooth family of Hamiltonian functions on M. This family generates a 1-parameter family of diffeomorphisms \varphi_t: M \to M by flowing along the Hamiltonian vector field associated to H_t. The Arnold–Givental conjecture states that if \varphi_1(L) intersects transversely with L, then :\# (\varphi_1(L) \cap L) \geq \sum_{i=0}^n \dim H_i(L; \mathbb Z / 2 \mathbb Z). Status The Arnold–Givental conjecture has been proved for several special cases. • Alexander Givental proved it for (M, L) = (\mathbb{CP}^n, \mathbb{RP}^n). • Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices. • Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number. • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for (M, \omega) semi-positive. • Urs Frauenfelder proved it in the case when (M, \omega) is a certain symplectic reduction, using gauged Floer theory. == See also ==