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Algebroid nature of the characteristic classes of flat bundles. (English) Zbl 0928.55018

Oprea, John (ed.) et al., Homotopy and geometry. Proceedings of the workshop, Banach Center, Warsaw, Poland, June 9–13, 1997. Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 45, 199-224 (1998).
The two homotopic notions of (a) homotopic homomorphisms between principal bundles and (b) homotopic subbundles are important in many domains of differential geometry. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that these notions can be, in a natural way, expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles are invariants of Lie algebroids of these bundles. This enables one to construct the characteristic homomorphism of a flat regular Lie algebroid, measuring the incompatibility of the flat structure with a given subalgebroid. For two given Lie subalgebroids, these homomorphisms are equivalent if the Lie subalgebroids are homotopic. Some new examples of applications of this characteristic homomorphism to a transitive case (for TC foliations) and to a non-transitive case (for a principal bundle equipped with a partial flat connection) are pointed out. An example of a transitive Lie algebroid of a TC-foliation which leads to a nontrivial characteristic homomorphism is obtained.
For the entire collection see [Zbl 0906.00019].

MSC:

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R20 Characteristic classes and numbers in differential topology
53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
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