Milnor Wood inequalities for hyperbolic manifolds

Tsachik Gelander (Jerusalem), mercredi 4 mars 2009, 14h15, salle MA12

Abstract:

In his 1958 celebrated paper, Milnor proved a classical inequality for the Euler number of flat vector bundles over surfaces. As a consequence it followed that closed surfaces with nonzero Euler characteristic admit no affine connection with 0 curvature, and in particular they are not affine. It is in fact an old conjecture that closed manifolds with nonzero Euler characteristic of any even dimension admits no flat structure, and the special case that such manifolds are not affine is often referred to as the Chern conjecture. The strong conjecture was shown to be false in general by Smillie, but could still be true under some homogeneity assumption. Few special cases of Chern conjecture where confirmed: Kostant and Sullivan proved that closed manifolds with nonzero Euler characteristic admit no complete affine structure, and Hirsch and Goldman proved the Chern conjecture for manifolds admitting an irreducible higher rank locally symmetric Riemannian structure. In general however very little is yet known.

We proved Milnor type inequality for mflds admitting a Riemannian structure locally isometric to H^n where H is the hyperbolic plane. This is a first sharp generalization of Milnor's inequality to manifolds of dimension >2. As a result the conjecture that such manifolds admits no flat affine connection is confirmed. Note that for product of surfaces it was not even known if affine structure exists. Over irreducible manifolds of higher rank, we showed that the flat vector bundles stand in one to one correspondence with the elements of some finite cohomology group. In particular, a Hilbert-Blumenthal modular manifold M of dimension 2n admits exactly 2^(nk+1) flat vector bundle with nonzero Euler number, where k=dim H^1(M,Z/2).

Joint work with Michelle Bucher.