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A geometric construction of the first Pontryagin class. (English) Zbl 0837.57022

Kauffman, Louis H. (ed.) et al., Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. Ser. Knots Everything. 3, 209-220 (1993).
Summary: A new obstruction theory for principal bundles is developed, which leads to an integer-valued formula for the first Pontryagin class of a bundle with compact structure group. A geometric representative of this class is given, in terms of a glueing problem for gerbes. The upshot is that there is a natural sheaf of bicategories over the base manifold. Analogous constructions are discussed for finite groups, leading to a proof of the reciprocity theorem of Segal and Witten.
For the entire collection see [Zbl 0818.00013].

MSC:

57R20 Characteristic classes and numbers in differential topology
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