**An inverse to the Poincaré conjecture**

The generalized Poincaré conjecture concerns an elementary geometric
or better topological question: How can one detected that a given space is a
sphere? The conjecture (which apart from dimension 3, where there is also an
anouncement of a serious proof, is a theorem) says that on the one hand one
has to check local conditions, i.e. the space looks locally like the euclidean space. On
the other hand one has to check two algebraic topological conditions: the space
has to by simply connected and has to have the homology of a sphere. All these
conditions will be explained. Recently - motivated by a question from
combinatorics - I inverted the Poincaré conjecture in a certain sense. I will
explain this inverse problem and give some results in low dimensions.

*Matthias Kreck*

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