Arnold–Givental conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of on the number of intersection points of with a Hamiltonian isotopic Lagrangian submanifold which intersects transversally.
Let be a smooth family of Hamiltonian functions of and denote by the time-one map of the flow of the Hamiltonian vector field of. Let be a Lagrangian submanifold, invariant under some antisymplectic involution of. Assume that and intersect transversally. Then the number of intersection points of and can be estimated from below by the sum of the Betti numbers of, i.e.
Up to now, the Arnold–Givental conjecture could only be proven under some additional assumptions.
