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Added to Maslov index and to Lagrangian Grassmannian the following quick cohomological definition of the Maslov index:
The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers
$H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,.$The generator of this cohomology group is called the universal Maslov index
$u \in H^1(LGrass, \mathbb{Z}) \,.$Given a Lagrangian submanifold $Y \hookrightarrow X$ of a symplectic manifold $(X,\omega)$, its tangent bundle is classified by a function
$i \;\colon\; Y \to LGrass \,.$The _Maslov index of $Y$ is the universal Maslov index pulled back along this map
$i^\ast u \in H^1(Y,\mathbb{Z}) \,.$1 to 3 of 3