Arnold's nearby Lagrangian conjecture

Theorem 1.1: for some sufficiently large even $N$, $Q=S^9\times S^{N-1}$ admits a closed exact smoothly embedded Lagrangian $L\subset T^*Q$, diffeomorphic to $Q$ and with $Q$ and $L$ simply connected, that no compactly supported Hamiltonian isotopy carries to the zero section. $L$ is cut out by a generating family whose data at infinity form a nontrivial class $a\in\pi_9$ of the stable smooth tube space with trivial associated spherical fibration; a Morse and h-cobordism argument shows such data cannot come from a family with one nondegenerate critical point per fibre, as an isotopy to the zero section would force. Not shown: an explicit $N$, counterexamples in low dimension, or any failure of the homotopy-theoretic consequences already proved.

Theorem 1.1: for some sufficiently large even $N$, $Q=S^9\times S^{N-1}$ admits a closed exact smoothly embedded Lagrangian $L\subset T^*Q$, diffeomorphic to $Q$ and with $Q$ and $L$ simply connected, that no compactly supported Hamiltonian isotopy carries to the zero section. $L$ is cut out by a generating family whose data at infinity form a nontrivial class $a\in\pi_9$ of the stable smooth tube space with trivial associated spherical fibration; a Morse and h-cobordism argument shows such data cannot come from a family with one nondegenerate critical point per fibre, as an isotopy to the zero section would force. Not shown: an explicit $N$, counterexamples in low dimension, or any failure of the homotopy-theoretic consequences already proved.