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Holonomy, twisting cochains and characteristic classes. (English) Zbl 1276.57028

Summary: 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 [J.-M. Bismut, C. R. Acad. Sci., Paris, Sér. I 297, 481–484 (1983; Zbl 0539.58034), E. Getzler, J. D. S. Jones and S. Petrack, Topology 30, No.3, 339–371 (1991; Zbl 0729.58004) and L. Q. Zamboni, Trans. Am. Math. Soc. 331, No.1, 157–163 (1992; Zbl 0762.55004)]), also called Bismut’s class. The latter example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to Witten’s index formula. We also give several examples of the twisting cochain associated with a given principal bundle. In particular, our approach allows us to obtain explicit formulas for the Chern classes and for an analogue of the cyclic Chern character in the terms of the glueing functions of the principal bundle. We discuss some modifications of this construction. We hope that this approach can turn fruitful for the investigations of the Witten index formula.

MSC:

57R22 Topology of vector bundles and fiber bundles
57R20 Characteristic classes and numbers in differential topology
57R19 Algebraic topology on manifolds and differential topology
55N99 Homology and cohomology theories in algebraic topology
53C29 Issues of holonomy in differential geometry
58J20 Index theory and related fixed-point theorems on manifolds
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