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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