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 $H t \in C \infty (M); 0 \leq t \leq 1$ be a smooth family of Hamiltonian functions of $M$ and denote by $φ H$ the time-one map of the flow of the Hamiltonian vector field $X H t$ of $H t$ . Let $L$ be a Lagrangian submanifold, invariant under some antisymplectic involution of $M$ . 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 $Z 2$ 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,^[when?] the Arnold–Givental conjecture could only be proven under some additional assumptions.

References[]

Frauenfelder, Urs (2004), "The Arnold–Givental conjecture and moment Floer homology", International Mathematics Research Notices, 2004 (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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Arnold–Givental conjecture

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