Homological Lagrangian monodromy (Q640343)

The authors show that the Hamiltonian Lagrangian monodromy group (see [\textit{M.-L. Yau}, Math. Res. Lett. 16, No.~2--3, 531--541 (2009; Zbl 1181.53071)]) is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The main result is the following: Let \((M,\omega)\) be a symplectic manifold and \(L\) a closed weakly exact Lagrangian submanifold. Let \(g_t\) be a Hamiltonian homotopy of \(M\) starting at the identity and ending at a diffeomorphism preserving \(L\). Let \(f = g_1|_L\), then the induced map on the homology \(f_* : H_* (L, \mathbb{Z}_2) \to (L, \mathbb{Z}_2)\) is the identity. The proof uses the sheaf of Floer homologies of a Lagrangian submanifold induced by some Lagrangian fibration and the relative Seidel morphism (see [\textit{S. Hu} and \textit{F. Lalonde}, Trans. Am. Math. Soc. 362, No.~3, 1135--1168 (2010; Zbl 1189.53076)]).