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.

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

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

See also

• Arnold conjecture

References

• Frauenfelder, Urs (2004), "The Arnold–Givental conjecture and moment Floer homology", International Mathematics Research Notices (42): 2179–2269, doi:10.1155/S1073792804133941, MR 2076142.
• Oh, Yong-Geun (1992), "Floer cohomology and Arnol'd-Givental's conjecture of [on] Lagrangian intersections", Comptes Rendus de l'Académie des Sciences, 315 (3): 309–314, MR 1179726.