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This paper contains a description of various geometric constructions associated with fibre bundles, given in terms of important algebraic object, the “twisting cochain". Our examples include the Chern-Weil classes, the holonomy representation and the so-called cyclic Chern character of Bismut and others (see [2, 11, 27]), also called the Bismut’s class. The later example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to the Witten’s index formula....
We employ the notion of the universal differential calculus to propose yet another approach to the Hopf-type cohomology of Hopf algebras.
We propose a new construction of characteristic classes for noncommutative algebraic principal bundles (Hopf-Galois extensions) with values in Hochschild and cyclic homology.
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