Colloquium Lecture Department of Mathematics Bar Ilan University

Sunday 10 May '09, at 12:00 pm Department Room

Prof. T. Gelander, Hebrew University, Jerusalem

Title: Flat manifolds and Euler numbers (joint work with M. Bucher)

Abstract: It is an old conjecture that aspherical (even dimensional) closed manifolds with nonzero Euler characteristic cannot admit flat structures. In 1958 Milnor proved it in dimension 2, but in higher dimension very little progress was made. In 1975 Hirsch and Thurston proved it for manifolds whose fundamental group is a free product of virtually solvable groups. Note that the well known Chern conjecture, that affine manifolds have zero Euler characteristic, is a particular case of the general conjecture. Jointly with Michelle Bucher we proved the general conjecture (and in particular Chern's conjecture) for manifolds that can be locally decomposed as a product of symmetric planes. Moreover we proved a sharp generalization of the Milnor-Wood inequality for such manifolds which immediately implies the conjecture. For Hilbert-Blumenthal modular varieties we also characterized all the flat vector bundles with nonzero Euler number, and showed that the list includes no tangent bundles.