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Summary:
The aim of this paper is to construct a natural mapping $\check C\sb k$, $k=1,2,3,\dots$, from the multiplicative $K$-theory $K(X)$ of a differential manifold $X$, associated to the trivial filtration of the de Rham complex, as defined by {\it M. Karoubi} in [C. R. Acad. Sci., Paris, S\'er. I 302, 321-324 (1986; Zbl 0593.55004)] to the odd cohomology $H\sb s\sp{2k-1} (X;C\sp*)$. By using this mapping, the author associates to any flat complex vector bundle $E$ on $X$ characteristic classes $\check C\sb k(E) \in H\sb{dR}\sp{2k-1} (X;C\sp*)$ analogous to the classes studied by {\it S. Chern}, {\it J. Cheeger} and {\it J. Simons} in [Differential characters and geometric invariants, in `Geometry and topology', Lect. Notes Math. 1167, 50-80 (1985; Zbl 0621.57010), Characteristic forms and geometric invariants, Ann. Math., II. Ser. 99, 48-69 (1974; Zbl 0283.53036)].
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