Comptes Rendus
no. 11-12
Topology/Geometry
Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes
[Inégalités de Milnor–Wood pour variétés localement isométriques à un produit de plans hyperboliques]

Présenté par : Étienne Ghys

Michelle Bucher1 ; Tsachik Gelander2
1 KTH Mathematics Department, 10044 Stockholm, Sweden
2 Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 661-666.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.04.014
Michelle Bucher 1 ; Tsachik Gelander 2

1 KTH Mathematics Department, 10044 Stockholm, Sweden
2 Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel 
@article{CRMATH_2008__346_11-12_661_0,
     author = {Michelle Bucher and Tsachik Gelander},
     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},
}
TY  - JOUR
AU  - Michelle Bucher
AU  - Tsachik Gelander
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
DO  - 10.1016/j.crma.2008.04.014
LA  - en
ID  - CRMATH_2008__346_11-12_661_0
ER  - 
%0 Journal Article
%A Michelle Bucher
%A Tsachik Gelander
%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
%R 10.1016/j.crma.2008.04.014
%G en
%F CRMATH_2008__346_11-12_661_0
Michelle Bucher; Tsachik Gelander. Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 661-666. doi : 10.1016/j.crma.2008.04.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.014/

[1] M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by H2×H2, J. Topology (2008), in press

[2] J.-L. Clerc; B. Ørsted The Gromov norm of the Kaehler class and the Maslov index, Asian J. Math., Volume 7 (2003) no. 2, pp. 269-295

[3] A. Domic; D. Toledo The Gromov norm of the Kaehler class of symmetric domains, Math. Ann., Volume 276 (1987) no. 3, pp. 425-432

[4] M. Gromov 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] N.V. Ivanov; V.G. Turaev A canonical cocycle for the Euler class of a flat vector bundle, Soviet Math. Dokl., Volume 26 (1982) no. 1, pp. 78-81

[7] B. Kostant; D. Sullivan The Euler characteristic of a compact affine space form is zero, Bull. Amer. Math. Soc., Volume 81 (1974), pp. 937-938

[8] J. Milnor On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223

[9] J. Smillie 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] J. Wood Bundles with totally disconnected structure group, Comment. Math. Helv., Volume 46 (1971), pp. 257-273

Cité par Sources :

Commentaires - Politique