Some remarks on the Maslov index

It is a classical fact that Wall's index of three Lagrangians in a symplectic space over a field k defines a 2-cocycle W on the associated symplectic group with values into the Witt group of k. Moreover, module the square of the fundamental ideal this is a trivial 2-cocycle. In this work we revisit this fact from the viewpoint of the theory of Sturm sequences and Sylvester matrices developed by J. ~Barge and J. ~Lannes in their book ``Suites de Sturm, indice de Maslov et periodicite de Bott. (French) Sturm sequences, Maslov index and Bott periodicity'' Progress in Mathematics, 267. This allows us in particular to give an explicit formula for the coboundary associated to the mod I² reduction of the cocycle W which is valid on any field of characteristic different from 2