# Jerusalem Mathematics Colloquium

Thursday, 26th March 2009, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Tsachik Gelander

(HU)

"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.

Light refreshments will be served in the faculty lounge at 3:30.

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