On a characteristic class arising in quantization conditions

Abstract

Recently V. P. Maslov gave a mathematically rigorous treatment of multidimensional asymptotic methods of ″quasiclassical″ type in the large, i.e., for any number of conjugate points [1, 2]. It turned out that there appeared in the asymptotic formulas certain integers, reflecting homological properties of curves on surfaces of the phase space and closely related to the Morse indexes of the corresponding variational problems. In particular, Maslov defined a one-dimensional class of integer-valued cohomologies whose values on the basis cycles enter into the so-called ″quantization conditions.″

In this note we give new formulas for the calculation of this class of cohomologies. This class is characteristic in the category of real vector bundles, whose complexification is trivial and trivialized, and also in certain wider categories.