Arnold conjecture

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Let be a compact symplectic manifold. For any smooth function , the symplectic form induces a Hamiltonian vector field on , defined by the identity The function is called a Hamiltonian function. Suppose there is a 1-parameter family of Hamiltonian functions , inducing a 1-parameter family of Hamiltonian vector fields on . The family of vector fields integrates to a 1-parameter family of diffeomorphisms . Each individual of is a Hamiltonian diffeomorphism of . The Arnold conjecture says that for each Hamiltonian diffeomorphism of , it possesses at least as many fixed points as a smooth function on possesses critical points. A Hamiltonian diffeomorphism is called nondegenerate if its graph intersects the diagonal of transversely. For nondegenerate Hamiltonian diffeomorphisms, a 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 , called the Morse number of . In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of , for example, the sum of Betti numbers over a field : The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on the above integer is a lower bound of its number of fixed points.