Vortragsveranstaltung

**The mystery of aspherical closed manifolds**

23.1.2008, 16:00 Uhr

A closed manifold is called aspherical if its universal
covering is contractible. Such objects occur very often
in interesting situations such as Riemannian manifolds
with non-positive sectional curvature or irreducible 3-manifolds.

We want to discuss certain rigidity properties. A famous example is the
Borel Conjecture which predicts that two closed aspherical manifolds are
homeomorphic if and only if their fundamental groups are isomorphic.
Another prominent conjecture is the generalization of the Hopf/Singer
Conjecture that for aspherical closed manifolds the Euler characteristic
satisfies a certain parity condition depending on the dimension and that
its
-Betti numbers are concentrated in the middle dimension.

The question which groups do occur as fundamental groups of closed
aspherical manifolds yields interesting connections to group homology.

Although some of these conjectures and questions have been solved in
many cases, there is no satisfying explanation why the condition
aspherical has all these consequences.

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