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In this paper we give explicit formulas for differential characteristic classes of principal
-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and the differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons–Sullivan) differential
-theory to (Cheeger–Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern class on Freed–Lott differential
-theory. Finally, we discuss the odd differential Chern classes.
We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.
Let (P,Y ) be a bundle gerbe over a fibre bundle Y →M. We show that if M is simply connected and the fibres of Y →M are connected and finite-dimensional, then the Dixmier–Douady class of (P,Y ) is torsion. This corrects and extends an earlier result of the first author.
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