A relative version of Weinstein's morphism

Abstract

We introduce a relative version of Weinstein's morphism associated to a Lagrangian submanifold \(L\) of a closed symplectic manifold \((M,ω)\). We construct a homomorphism \[ \mathcal{A}^{(L,α)}\colon π_{1}\bigl(\operatorname{Ham}_0(M,L)\bigr) \longrightarrow \mathbb{R}/\mathcal{P}_{2}(M,ω), \] and establish its basic properties. We compute the invariant in several examples, including latitude circles in \(S^2\), and more generally, Lagrangian fibers in toric manifolds. These computations provide obstructions to Ha\-mil\-to\-nian isotopy of Lagrangian submanifolds and detect nontrivial, and in some cases infinite-order, elements in the homotopy groups of their Hamiltonian stabilizers.