Algebroid nature of the characteristic classes of flat bundles

Jan Kubarski

Banach Center Publications (1998)

  • Volume: 45, Issue: 1, page 199-224
  • ISSN: 0137-6934

Abstract

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The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat 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 (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation which leads to the nontrivial characteristic homomorphism is obtained.

How to cite

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Kubarski, Jan. "Algebroid nature of the characteristic classes of flat bundles." Banach Center Publications 45.1 (1998): 199-224. <http://eudml.org/doc/208904>.

@article{Kubarski1998,
abstract = {The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat 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 (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation which leads to the nontrivial characteristic homomorphism is obtained.},
author = {Kubarski, Jan},
journal = {Banach Center Publications},
keywords = {principal bundles; characteristic classes; Lie algebroids; TC-foliation; flat bundles},
language = {eng},
number = {1},
pages = {199-224},
title = {Algebroid nature of the characteristic classes of flat bundles},
url = {http://eudml.org/doc/208904},
volume = {45},
year = {1998},
}

TY - JOUR
AU - Kubarski, Jan
TI - Algebroid nature of the characteristic classes of flat bundles
JO - Banach Center Publications
PY - 1998
VL - 45
IS - 1
SP - 199
EP - 224
AB - The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat 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 (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation which leads to the nontrivial characteristic homomorphism is obtained.
LA - eng
KW - principal bundles; characteristic classes; Lie algebroids; TC-foliation; flat bundles
UR - http://eudml.org/doc/208904
ER -

References

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