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 Ht ∈ C ∞(M); 0 ≤ t ≤ 1 be a smooth family of Hamiltonian functions of M and denote by φH the time-one map of the flow of the Hamiltonian vector field XHt of Ht. Let L be a Lagrangian submanifold, invariant under some antisymplectic involution of M. Assume that L and φH (L) intersect transversally. Then the number of intersection points of L and φH (L) can be estimated from below by the sum of the Z2 Betti numbers of L, i.e.

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, arXiv:math/0309373, 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.
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