Some remarks on the Maslov index

TL;DR

This paper revisits Wall's Maslov index cocycle for Lagrangians in symplectic spaces, refining it using Sturm sequences and Sylvester matrices to provide explicit formulas and computations over fields with characteristic not 2.

Contribution

It introduces a refined cocycle related to Wall's index, utilizing Sylvester matrices to derive explicit formulas and calculations over various fields.

Findings

Defined a 2-fold refinement of Wall's cocycle.

Provided explicit formulas for the coboundary in terms of Sylvester matrices.

Computed coboundary values on standard symplectic group elements.

Abstract

It is a classical fact that Wall's index of a triplet of Lagrangians in a symplectic space over a field $k$ defines a $2$-cocycle $\mu_W$ on the associated symplectic group with values in the Witt group of $k$. Moreover, modulo 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 teir book Suites de Sturm, indice de Maslov et p\'eriodicit\'e de Bott, volume 267 of Progress in Mathematics. Birkh\"auser Verlag, Basel, 2008. We define a refinement by a factor of $2$ of Wall's cocycle and use the technology of Sylvester matrices to give an explicit formula for the coboundary associated to the mod $I^2$ reduction of the cocycle which is valid for any field of characteristic different from $2$. Finally we explicitly compute the values of the coboundary on standard elements of the symplectic group.