Maslov index

Summary

The Maslov index is an integer attached to a loop of Lagrangian subspaces of the symplectic vector space $\R^{2n}$. It is the class of the loop in the fundamental group of the Lagrangian Grassmannian, which is isomorphic to $\Z$. It can be applied to the tangent planes of a Lagrangian submanifold along a loop. Then it counts with signs how often the loop crosses the Lagrangian singularities of the projection to the base. It enters the quantization rules of semiclassical mechanics, the index theory of Hamiltonian systems, and Floer homology.