A Cocycle for the Symplectic First Chern Class and the Maslov Index

Let G ~ Sp (n, R) be the symplectic group (n~i) and G 6 be the same group with the discrete topology. The present paper is devoted to the study of the element u of the group H ~ (G~; Z), which is determined up to sign in any of the following (equivalent) ways: a) u is the image under the canonical homomorphism H 2 (BG; Z)-~ H 2 (BG~: Z) of the generator of the group H 2 (BG; Z)~ Z; b# u is the cohomology class corresponding to an extension of the group G by its universal covering, provided with the natural group structure (recall that ~I(G) ~ Z); c) u is the first Chern class of the complex vector bundle obtained from the real vector bundle over BG~(~ K (G6,1)) associated with the universal principal G6-bundle and the action of G on R 2~ introduced by the natural complex structure (see [2, 3]).