Holonomy, twisting cochains and characteristic classes

Abstract

The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a 1-connected manifold. We regard it as a homomorphism from the algebra of differential forms on the structure group to the algebra of differential forms on the based loopspace of the base, represented by the (reduced) bar-complex of differential forms on it. Next we discuss the notion of "twisting cochains", or more generally "twisting maps", their equivalence relation and give various examples. We show that every twisting map gives rise to a map from the coalgebra to the bar-resolution of the algebra. Further we show that in the case of genuine twisting cochains one can obtain a map from the differential forms on the gauge bundle, associated with the given principal one, to the reduced Hochschild complex of the algebra, of differential forms of the base. Then we discuss a concrete example of a twisting cochain that is defined on the polynomial de Rham forms on an algebraic group and takes values in Cech complex of the base. We show how it can be used to obtain explicit formulas for the Chern classes. We also discuss few modifications of this construction. In the last section we discuss the construction, similar to the one, used by Getzler, Jones and Petrack in their 1991 paper. We show that the map we call "Getzler-Jones-Petrack's map" is homotopy-equivalent to the map that one obtains from a twisting cochain. This enables us to find a generalization of the Bismut's class, which we regard as an image of a suitable element in the differential forms on the group under the Getzler-Jones-Petrack's map.