In this paper we give four definitions of Maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other. Moreover, relationships of several symplectic and differential geometric, analytic, and topological invariants (including triple Maslov indices, eta invariants, spectral flow and signatures of quadratic forms) to the Maslov index are developed and formulae relating them are given. The broad presentation is designed with a view to applications both in geometry and in analysis. IntroductionThe object of this paper is to give a systematic and unified treatment of the Maslov index and some related invariants. In the literature, the Maslov index has often been described as an integer invariant associated to any one of the following situations: Here all three will be considered and compared with each other in Sections 1 to 9. Following [5] and [12], we regard the setting (i) as the main theme, whereas the others are variations.Let ( V , { , }) be a fixed symplectic vector space with symplectic pairing {,}, and let Lag(V) be the space of Lagrangian subspaces in V.

This paper is the first of a three-part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self-adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L2-solutions on both sides and a common limiting value of two extended L2-solutions. In Part 11, the present analytic "Mayer-Vietoris" results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part I11 after comparing infinite-and finite-dimensional Lagrangians and determinant line bundles and then introducing "canonical perturbations" of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3-manifolds, including Casson's invariant.

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