### Topological rigidity for non-aspherical manifolds, by Matthias Kreck and Wolfgang Lueck

The Borel Conjecture predicts that closed aspherical manifolds are
topological rigid. We want to investigate when a non-aspherical oriented
connected closed manifold M is topological rigid in the following sense. If f:
N ---> M is an orientation preserving homotopy equivalence with a closed
oriented manifold as target, then there is an orientation preserving
homeomorphism h: N ---> M such that h and f induce up to conjugation the same
maps on the fundamental groups. We call such manifolds Borel manifolds. We give
partial answers to this questions for S^k x S^d, for sphere bundles over
aspherical closed manifolds of dimension less or equal to 3 and for 3-manifolds
with torsionfree fundamental groups. We show that this rigidity is inherited
under connected sums in dimensions greater or equal to 5. We also classify
manifolds of dimension 5 or 6 whose fundamental group is the one of a
surface and whose second homotopy group is trivial.

Matthias Kreck <kreck@mathi.uni-heidelberg.de>

Wolfgang Lueck <lueck@math.uni-muenster.de>