# Algebroid nature of the characteristic classes of flat bundles

Banach Center Publications (1998)

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

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topKubarski, 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

top- [A-M] R. Almeida and P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985).
- [B] B. Balcerzak, Lie algebroid of a vector bundle, in preparation. Zbl0967.58012
- [C-D-W] A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, Publ. Dep. Math. Université de Lyon 1, 2/A (1987).
- [G] W. Greub, Multilinear algebra, Springer-Verlag, 1967.
- [G-H-V] W. Greub, S. Halperin and R. Vanstone, Connections, Curvature, and Cohomology, Vol. II, 1973, Vol. III, 1976. Zbl0335.57001
- [H-M] Ph. Higgins and K. Mackenzie, Algebraic construction in the category of Lie algebroids, Journal of Algebra 129 (1990), 194-230. Zbl0696.22007
- [K-T] F. Kamber and Ph. Tondeur, Foliated Bundles and Characteristic Classes, Lectures Notes in Mathematics 493, Springer-Verlag 1975. Zbl0308.57011
- [K1] J. Kubarski, Pradines-type groupoids over foliations; cohomology, connections and the Chern-Weil homomorphism, Technical University, Institute of Mathematics, Preprint 2, August 1986.
- [K2] J. Kubarski, Exponential mapping for Lie groupoids. Applications, Colloquium Mathematicum, 54 (1987), 39-48. Zbl0638.22001
- [K3] J. Kubarski, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. Université de Lyon 1, 1/A, 1989.
- [K4] J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. Université de Lyon 1, 1991.
- [K5] J. Kubarski, A criterion for the Minimal Closedness of the Lie Subalgebra Corresponding to a Connected Nonclosed Lie Subgroup, Revista Matematica de la Universidad Complutense de Madrid 4, 1991. Zbl0766.17004
- [K6] J. Kubarski, Tangential Chern-Weil homomorphism, Proceedings of Geometric Study of Foliations, Tokyo, Nov. 1993, World Scientific, Singapore, 1994, pp. 324-344.
- [K7] J. Kubarski, Invariant cohomology of regular Lie algebroids, Proceedings of the VII International Colloquium on Differential Geometry, 'Analysis and Geometry in Foliated Manifolds', Santiago de Compostela, Spain 26-30 July, 1994, World Scientific, Singapore 1995.
- [K8] J. Kubarski, Fibre integral in regular Lie algebroids, Proceedings of the Conference 'Differential Geometry', Budapest, July 27-30, 1996, to appear.
- [K9] J. Kubarski, Some properties of regular Lie algebroids over foliated manifolds, in preparation.
- [KU] A. Kumpera, An introduction to Lie groupoids, Nucleo de Estudos e Pasquisas Cientificas, Rio de Janeiro, 1971.
- [M1] K. Mackenzie, Lie groupoids and Lie algebroids in Differential Geometry, London Mathematical Society Lecture Note Series 124, Cambridge, 1987. Zbl0683.53029
- [MR] L. Maxim-Raileanu, Cohomology of Lie algebroids, An. Sti. Univ. 'Al. I. Cuza' Iasi. Sect. I a Mat. 22 (1976), 197-199.
- [M-S] C. Moore and C. Schochet, Global Analysis on Foliated Spaces, Mathematical Sciences Research Institute publications 9, Springer-Verlag, 1988. Zbl0648.58034
- [MO1] P. Molino, Etude des feuilletages transversalement complets et applications, Ann. Sci. Ecole Norm. Sup. 10(3) (1977), 289-307. Zbl0368.57007
- [MO2] P. Molino, Riemannian foliations, Progress in Mathematics Vol. 73, Birkhäuser, Boston, Basel, 1988.
- [N-V-Q] Ngô Van Quê, Du prolongement des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier (Grenoble) 17,1 (1967), 157-223. Zbl0157.28506
- [P1] J. Pradines, Théorie de Lie pour les groupoï des différentiables. Calcul différentiel dans la catégorie des groupoï des infinitésimaux, C. R. Acad. Sci. Ser. A-B, Paris 264 (1967), 245-248. Zbl0154.21704
- [P2] J. Pradines, Théorie de Lie pour les groupoï des différentiables, Atti Conv. Intern. Geom. 7 Diff. Bologna, 1967, Bologna-Amsterdam.

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