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 Template:Mvar on the number of intersection points of Template:Mvar with a Hamiltonian isotopic Lagrangian submanifold which intersects Template:Mvar transversally.

Let Template:Math be a smooth family of Hamiltonian functions of Template:Mvar and denote by Template:Mvar the one-time map of the flow of the Hamiltonian vector field Template:Mvar of Template:Mvar. Let Template:Mvar be a Lagrangian submanifold, invariant under some antisymplectic involution of Template:Mvar. Assume that Template:Mvar and Template:Math intersect transversally. Then the number of intersection points of Template:Mvar and Template:Math can be estimated from below by the sum of the Template:Math Betti numbers of Template:Mvar, i.e.

$\left|L\cap \varphi _{H}(L)\right|\geq \sum _{k=0}^{n}b_{k}\left(L;\mathbf {Z} _{2}\right)$

Up to now,Template:When the Arnold–Givental conjecture could only be proven under some additional assumptions.

See also

• Arnold conjecture

References

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