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

Let H_t ∈ C ^∞(M); 0 ≤ t ≤ 1 be a smooth family of Hamiltonian functions of M and denote by φ_H the one-time map of the flow of the Hamiltonian vector field X_Ht of H_t. Assume that L and φ_H (L) intersect transversally. Then the number of intersection points of L and φ_H (L) can be estimated from below by the sum of the Z₂ Betti numbers of L, 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, the Arnold–Givental conjecture could only be proven under some additional assumptions.

See also

• Arnold conjecture

References

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