[Inégalités de Milnor–Wood pour variétés localement isométriques à un produit de plans hyperboliques]
Nous généralisons l'inégalité classique de Milnor aux variétés localement isométriques à un produit de plans hyperboliques. Il en découle que de telles variétés n'admettent pas de structure affine, confirmant dans ce cas la conjecture de Chern–Sullivan. Contrairement à de nombreuses variétés localement symétriques, les variétés considérées dans cette Note admettent un fibré vectoriel plat en dimension correspondante. Si les variétés sont de plus irréductibles de rang supérieur, nous montrons qu'un fibré vectoriel orienté plat avec nombre d'Euler non nul est, à orientation près, unique.
This Note describes sharp Milnor–Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern–Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to some other locally symmetric spaces they do admit interesting flat vector bundles in the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number.
Accepté le :
Publié le :
@article{CRMATH_2008__346_11-12_661_0,
author = {Bucher, Michelle and Gelander, Tsachik},
title = {Milnor{\textendash}Wood inequalities for manifolds locally isometric to a product of hyperbolic planes},
journal = {Comptes Rendus. Math\'ematique},
pages = {661--666},
publisher = {Elsevier},
volume = {346},
number = {11-12},
year = {2008},
doi = {10.1016/j.crma.2008.04.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.04.014/}
}
TY - JOUR AU - Bucher, Michelle AU - Gelander, Tsachik TI - Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes JO - Comptes Rendus. Mathématique PY - 2008 SP - 661 EP - 666 VL - 346 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.04.014/ DO - 10.1016/j.crma.2008.04.014 LA - en ID - CRMATH_2008__346_11-12_661_0 ER -
%0 Journal Article %A Bucher, Michelle %A Gelander, Tsachik %T Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes %J Comptes Rendus. Mathématique %D 2008 %P 661-666 %V 346 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.04.014/ %R 10.1016/j.crma.2008.04.014 %G en %F CRMATH_2008__346_11-12_661_0
Bucher, Michelle; Gelander, Tsachik. Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 661-666. doi : 10.1016/j.crma.2008.04.014. http://www.numdam.org/articles/10.1016/j.crma.2008.04.014/
[1] M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by , J. Topology (2008), in press
[2] The Gromov norm of the Kaehler class and the Maslov index, Asian J. Math., Volume 7 (2003) no. 2, pp. 269-295
[3] The Gromov norm of the Kaehler class of symmetric domains, Math. Ann., Volume 276 (1987) no. 3, pp. 425-432
[4] Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math., Volume 56 (1982), pp. 5-99
[5] F. Hirzebruch, Automorphe Formen und der Satz von Riemann–Roch, in: Symposium Internacional de topologia alebraics, Universidad Nacional Autonoma de Mexico y UNESCO, 1958
[6] A canonical cocycle for the Euler class of a flat vector bundle, Soviet Math. Dokl., Volume 26 (1982) no. 1, pp. 78-81
[7] The Euler characteristic of a compact affine space form is zero, Bull. Amer. Math. Soc., Volume 81 (1974), pp. 937-938
[8] On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223
[9] Flat manifolds with non-zero Euler characteristics, Comment. Math. Helv., Volume 52 (1977), pp. 453-455
[10] W. Thurston, Geometry and topology of 3-manifolds, Lecture Notes, Princeton, 1978
[11] Bundles with totally disconnected structure group, Comment. Math. Helv., Volume 46 (1971), pp. 257-273
Cité par Sources :
