Lagrangian intersection under Legendrian deformations (Q1816476)

From Arnold's conjecture for Lagrangian submanifolds -- ``Let \(L\) be a Lagrangian submanifold in a symplectic manifold \((M,\omega)\). For a Hamiltonian isotopy \(\{\phi_t|0\leq t\leq 1\}\), the number of intersection points of \(L\) and \(\phi_1(L)\) is at least the cup-length of \(L\). Moreover, if they intersect transversally, this number is at least the sum of Betti numbers of \(L\). ``This paper deals with the following particular question: ``Let \(\pi:P\to M\) be a principal \(S^1\)-bundle of \((M,\omega)\), with a connection such that the curvature equals \(-\omega\) and \(\Lambda\) a Legendrian submanifold in \(P\). Does the number of intersection points of \(L\) and \(\pi\circ \psi_1(\Lambda)\) satisfy the same inequality as above? \((\{\psi_t\mid 0\leq t\leq 1\}\) is a contact isotopy group on \(P\)).'' The author shows that this number is at least the sum of \(\mathbb{Z}_2\)-Betti numbers of \(L\), provided that \(\pi_2(M,L)=0\), and \(L\) and \(\pi\circ \psi_1(\Lambda)\) intersect transversally.