Arnold–Givental conjecture

Statement

Let ${\displaystyle (M,\omega )}$ be a compact ${\displaystyle 2n}$-dimensional symplectic manifold, let ${\displaystyle L\subset M}$ be a compact Lagrangian submanifold of ${\displaystyle M}$, and let ${\displaystyle \tau :M\to M}$ be an anti-symplectic involution, that is, a diffeomorphism ${\displaystyle \tau :M\to M}$ such that ${\displaystyle {\tau }^{\ast }\omega =-\omega }$ and ${\displaystyle {\tau }^2={\text{id}}_M}$, whose fixed point set is ${\displaystyle L}$.

Let ${\displaystyle H_t\in C^{\mathrm{\infty }}(M)}$, ${\displaystyle t\in [0,1]}$ be a smooth family of Hamiltonian functions on ${\displaystyle M}$. This family generates a 1-parameter family of diffeomorphisms ${\displaystyle {\phi }_t:M\to M}$ by flowing along the Hamiltonian vector field associated to ${\displaystyle H_t}$. The Arnold–Givental conjecture states that if ${\displaystyle {\phi }_1(L)}$ intersects transversely with ${\displaystyle L}$, then

${\displaystyle \mathrm{\#}({\phi }_1(L)\cap L)\ge \sum _{i=0}^n\dim H_i(L;{\mathbb{Z}}_2)}$.

One version of the Arnold conjecture can be obtained from the Arnold–Givental conjecture by considering the diagonal ${\displaystyle \mathrm{\Delta }}$ as a Lagrangian submanifold of ${\displaystyle (M\times M,\omega \oplus -\omega )}$ for ${\displaystyle M}$ a compact symplectic manifold and ${\displaystyle \tau :M\to M}$ as the anti-symplectic involution switching the two factors of ${\displaystyle M\times M}$.