"Flat manifolds and Euler numbers"
Abstract:It is an old conjecture that an even dimensional closed manifold with nonzero Euler characteristic cannot admit a flat structure. In 1958 Milnor proved it in dimension 2. Surprisingly in 1977 Smillie constructed a 4 dimensional flat manifold with nonzero Euler characteristic. However, it is still believed that the conjecture is true under some homogeneity assumptions, yet until recently no progress was made for dimension >2. A particular case is the well known Chern conjecture stating that affine manifolds have zero Euler characteristic. The Chern conjecture was confirmed in few special cases, notably by Kostant and Sullivan and by Goldman and Hirsch, but the general case is yet untouchable.
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 (e.g. product of surfaces or closed Hilbert Blumenthal varieties). In fact we proved a sharp generalization of the Milnor-Wood inequality for such manifolds which immediately implies the nonexistence of flat structure (when the Euler char doesn't vanish). For irreducible manifolds this inequality yields a complete list of the flat vector bundles with nonzero Euler number, and this list includes no tangent bundles.